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Formulario y Tablas de Probabilidad para los Cursos de Probabilidad, Inferencia Estad´ıstica y Econometr´ıa∗ Ernesto Barrios Zamudio1 ´ José Angel Garc´ıa Pérez2 Departamento Académico de Estad´ıstica Instituto Tecnol´ ogico Aut´ onomo de M´ exico

Octubre 2009 Versi´ on 1.00

Índice 2

1. Formulario 1.1. Variables Aleatorias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2. Distribuciones de Probabilidad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3. Distribuciones Bivariadas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4. Distribución Normal Bivariada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2. Tablas de Probabilidad

6

2.1. Distribución Binomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2. Distribución Poisson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

2.3. Distribución Normal Estándar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

2.4. Distribución Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

2.5. Distribución t de Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

2

2.6. Distribución χ Ji-Cuadrada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

2.7. Distribución F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.8. Distribución del estad´ıstico d de Durbin-Watson . . . . . . . . . . . . . . . . . . . . . . . . .

24

3. Tabla de N´ umeros Aleatorios

33

3.1. 1050 N´ umeros Seudoaleatorios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗

Material tomado de los documentos de trabajo DE-A09.1 y DE-A09.3, de los mismos autores.

1

[email protected]

2

1

33

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

1.

2

Formulario

1.1.

Variables Aleatorias

• Valor Esperado de g(X)  X  g(x)P (X = x), si X es discreta     x E(g(X)) = Z ∞     g(x)fX (x)dx si X es continua  −∞

• Propiedades de la funci´ on generadora de momentos MX+a (t) MbX (t) M X+a (t) b

= eat MX (t) = MX (bt) a t = e b t MX ( ) b

• Tercero y Cuarto Momentos con respecto a la Media E[(X − µ)3 ] = E(X 3 ) − 3E(X)E(X 2 ) + 2(E(X))3 E[(X − µ)4 ] = E(X 4 ) − 4E(X)E(X 3 ) + 6(E(X))2 E(X 2 ) − 3(E(X))4 • Coeficientes de Asimetr´ıa y de Kurtosis CA CK

= α3 =

µ3 3/2

µ2 µ4 = α4 = 2 µ2

• M´ etodo de Transformaci´ on de Variables Sea U = h(Y ), con h monótona creciente o decreciente en y. Entonces dy fU (u) = fY (y) , donde y = h−1 (u) du

c E. Barrios y J. A. Garc´ıa

v.1.00

Distribuciones de Probabilidad

Distribución

Uniforme discreta

∗

Notación

Soporte RX

unif{x1 , . . . , xK } x ∈ {x1 , . . . , xK }

Funci´ on de Probabilidad

E(X)

Var(X)

1 K

K 1 X xi K i=1

K 1 X (xi − E(X))2 K i=1

Funci´ on Generadora de Momentos 1 k

P

etxi

i

Bernoulli

Be(p)

x ∈ {0, 1}

px (1 − p)1−x

p

p(1 − p)

pet + (1 − p)

Binomial

Bin(n, p)

x ∈ {0, 1, . . . , n}

n x p (1 − p)n−x x

np

np(1 − p)

[pet + (1 − p)]n

Poisson

Po(λ)

x ∈ {0, 1, 2, . . .}

λx e−λ x!

λ

λ

Uniforme continua

unif(a, b)

a≤x≤b

1 b−a

a+b 2

(b − a)2 12

Normal

N(µ, σ 2 )

−∞ < x < ∞

µ

σ2

Gama∗

Gama(α, β)

x ∈ R+

αβ

αβ 2

√1 σ 2π

n exp − 12

x−µ 2 σ

xα−1 e−x/β Γ(α)β α

o

eλ(e

t

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

1.2.

−1)

etb − eta t(b − a) 1

eµt+ 2 σ

2 2

t

(1 − βt)−α

Notas: • Γ(α) =

Z

∞

uα−1 e−u du.

0

Entonces:

Γ(α + 1) = α · Γ(α);

Γ(1/2) =

√ π;

Γ(1) = 1;

Γ(n + 1) = n!, para n = 1, 2, . . . .

• Distribución exponencial: X ∼ Exp(λ). Entonces, X ∼ Γ(1, 1/λ)

y E(Y ) = n. 3

v.1.00

• Distribución Ji-cuadrada: Y ∼ χ2n . Entonces, Y ∼ Γ(n/2, 2/λ)

y E(X) = 1/λ.

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

1.3.

4

Distribuciones Bivariadas

• Funci´ on de Densidad Condicional f (x2 |x1 ) =

fX1 ,X2 (x1 , x2 ) fX1 (x1 )

• Valor Esperado de g(X1 , X2 )

E[g(X1 , X2 )] =

 XX    g(x1 , x2 )P (X1 = x1 , X2 = x2 )  x

x

Z1 Z 2     g(x1 , x2 )fX1 ,X2 (x1 , x2 )dx1 dx2

• Funci´ on Generadora de Momentos Conjunta MX1 ,X2 (t1 , t2 ) = E(et1 X1 +t2 X2 ) • Covarianza y Coeficiente de Correlaci´ on σ12 = Cov(X1 , X2 ) = E [(X1 − E(X1 ))(X2 − E(X2 ))] = E(X1 X2 ) − E(X1 )E(X2 ) ρX1 X2 =

σ12 σ1 σ2

• M´ etodo de Transformaci´ on de Variables Sean las variables aleatorias Y1 y Y2 funciones de las variables aleatorias X1 y X2 , de manera que las ecuaciones en y1 y y2 tienen soluci´ on u ńica para x1 y x2 en términos de y1 y y2 . Esto es, Y1 = g1 (X1 , X2 )

y

Y2 = g2 (X1 , X2 ) Si las funciones y1 y y2 tienen derivadas Jacobiano J(x1 , x2 ) = Entonces,

x1 = h1 (y1 , y2 ) x2 = h2 (y1 , y2 )

parciales continuas en todos los puntos (x1 , x2 ), y el determinante ∂g1 ∂x1 ∂g2 ∂x1

∂g1 ∂x2 ∂g2 ∂x2

6= 0,

para todo (x1 , x2 )

fY1 ,Y2 (y1 , y2 ) = fX1 ,X2 (h1 (y1 , y2 ), h2 (y1 , y2 )) |J (h1 (y1 , y2 ), h2 (y1 , y2 ))|−1 .

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

1.4.

5

Distribuci´ on Normal Bivariada

• Funci´ on de Densidad Conjunta " ( 2 2 #) 1 x1 − µ1 x2 − µ2 x1 − µ1 x1 − µ1 1 p fX1 ,X2 (x1 , x2 ) = + − 2ρ exp − 2(1 − ρ2 ) σ1 σ1 σ2 σ1 2πσ1 σ2 1 − ρ2 • Funci´ on Generadora de Momentos Conjunta 1 2 2 2 2 MX1 ,X2 (t1 , t2 ) = exp (t1 µ1 + t2 µ2 ) + σ t + 2ρσ1 σ2 t1 t2 + σ2 t2 2 1 1 • Valor Esperado y Varianza Condicionales σ2 (x1 − µ1 ) σ1 Var(X2 |X1 = x1 ) = σ22 (1 − ρ2 ) E(X2 |X1 = x1 ) = µ2 + ρ

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.

6

Tablas de Probabilidad

2.1.

Distribuci´ on Binomial X ∼ Binom(n, θ)

p = P (X ≤ x) =

x X n

k=0

k

α

p

θk (1 − θ)n−k = 1 − α

0

x

n

Tabla 2A. Probabilidades acumuladas p de la distribuci´ on binomial (n = 5, 6, 7, 8, 9). x 0 1 2 3 4 5

0.01 0.951 0.999 1.000 1.000 1.000 1.000

0.05 0.774 0.977 0.999 1.000 1.000 1.000

0.1 0.590 0.919 0.991 1.000 1.000 1.000

0.2 0.328 0.737 0.942 0.993 1.000 1.000

0.25 0.237 0.633 0.896 0.984 0.999 1.000

0.3 0.168 0.528 0.837 0.969 0.998 1.000

0.4 0.078 0.337 0.683 0.913 0.990 1.000

θ 0.5 0.031 0.188 0.500 0.813 0.969 1.000

0.6 0.010 0.087 0.317 0.663 0.922 1.000

0.7 0.002 0.031 0.163 0.472 0.832 1.000

0.75 0.001 0.016 0.104 0.367 0.763 1.000

0.8 0.000 0.007 0.058 0.263 0.672 1.000

0.9 0.000 0.000 0.009 0.081 0.410 1.000

0.95 0.000 0.000 0.001 0.023 0.226 1.000

0.99 0.000 0.000 0.000 0.001 0.049 1.000

n=6

0 1 2 3 4 5 6

0.941 0.999 1.000 1.000 1.000 1.000 1.000

0.735 0.967 0.998 1.000 1.000 1.000 1.000

0.531 0.886 0.984 0.999 1.000 1.000 1.000

0.262 0.655 0.901 0.983 0.998 1.000 1.000

0.178 0.534 0.831 0.962 0.995 1.000 1.000

0.118 0.420 0.744 0.930 0.989 0.999 1.000

0.047 0.233 0.544 0.821 0.959 0.996 1.000

0.016 0.109 0.344 0.656 0.891 0.984 1.000

0.004 0.041 0.179 0.456 0.767 0.953 1.000

0.001 0.011 0.070 0.256 0.580 0.882 1.000

0.000 0.005 0.038 0.169 0.466 0.822 1.000

0.000 0.002 0.017 0.099 0.345 0.738 1.000

0.000 0.000 0.001 0.016 0.114 0.469 1.000

0.000 0.000 0.000 0.002 0.033 0.265 1.000

0.000 0.000 0.000 0.000 0.001 0.059 1.000

n=7

0 1 2 3 4 5 6 7

0.932 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.698 0.956 0.996 1.000 1.000 1.000 1.000 1.000

0.478 0.850 0.974 0.997 1.000 1.000 1.000 1.000

0.210 0.577 0.852 0.967 0.995 1.000 1.000 1.000

0.133 0.445 0.756 0.929 0.987 0.999 1.000 1.000

0.082 0.329 0.647 0.874 0.971 0.996 1.000 1.000

0.028 0.159 0.420 0.710 0.904 0.981 0.998 1.000

0.008 0.063 0.227 0.500 0.773 0.938 0.992 1.000

0.002 0.019 0.096 0.290 0.580 0.841 0.972 1.000

0.000 0.004 0.029 0.126 0.353 0.671 0.918 1.000

0.000 0.001 0.013 0.071 0.244 0.555 0.867 1.000

0.000 0.000 0.005 0.033 0.148 0.423 0.790 1.000

0.000 0.000 0.000 0.003 0.026 0.150 0.522 1.000

0.000 0.000 0.000 0.000 0.004 0.044 0.302 1.000

0.000 0.000 0.000 0.000 0.000 0.002 0.068 1.000

n=8

0 1 2 3 4 5 6 7 8

0.923 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.663 0.943 0.994 1.000 1.000 1.000 1.000 1.000 1.000

0.430 0.813 0.962 0.995 1.000 1.000 1.000 1.000 1.000

0.168 0.503 0.797 0.944 0.990 0.999 1.000 1.000 1.000

0.100 0.367 0.679 0.886 0.973 0.996 1.000 1.000 1.000

0.058 0.255 0.552 0.806 0.942 0.989 0.999 1.000 1.000

0.017 0.106 0.315 0.594 0.826 0.950 0.991 0.999 1.000

0.004 0.035 0.145 0.363 0.637 0.855 0.965 0.996 1.000

0.001 0.009 0.050 0.174 0.406 0.685 0.894 0.983 1.000

0.000 0.001 0.011 0.058 0.194 0.448 0.745 0.942 1.000

0.000 0.000 0.004 0.027 0.114 0.321 0.633 0.900 1.000

0.000 0.000 0.001 0.010 0.056 0.203 0.497 0.832 1.000

0.000 0.000 0.000 0.000 0.005 0.038 0.187 0.570 1.000

0.000 0.000 0.000 0.000 0.000 0.006 0.057 0.337 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.077 1.000

n=9

0 1 2 3 4 5 6 7 8 9

0.914 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.630 0.929 0.992 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.387 0.775 0.947 0.992 0.999 1.000 1.000 1.000 1.000 1.000

0.134 0.436 0.738 0.914 0.980 0.997 1.000 1.000 1.000 1.000

0.075 0.300 0.601 0.834 0.951 0.990 0.999 1.000 1.000 1.000

0.040 0.196 0.463 0.730 0.901 0.975 0.996 1.000 1.000 1.000

0.010 0.071 0.232 0.483 0.733 0.901 0.975 0.996 1.000 1.000

0.002 0.020 0.090 0.254 0.500 0.746 0.910 0.980 0.998 1.000

0.000 0.004 0.025 0.099 0.267 0.517 0.768 0.929 0.990 1.000

0.000 0.000 0.004 0.025 0.099 0.270 0.537 0.804 0.960 1.000

0.000 0.000 0.001 0.010 0.049 0.166 0.399 0.700 0.925 1.000

0.000 0.000 0.000 0.003 0.020 0.086 0.262 0.564 0.866 1.000

0.000 0.000 0.000 0.000 0.001 0.008 0.053 0.225 0.613 1.000

0.000 0.000 0.000 0.000 0.000 0.001 0.008 0.071 0.370 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.086 1.000

n=5

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

7

Tabla 2B. Probabilidades acumuladas p de la distribuci´ on binomial (n = 10, 11, 12, 13, 14). x 0 1 2 3 4 5 6 7 8 9 10

0.01 0.904 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.05 0.599 0.914 0.988 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.1 0.349 0.736 0.930 0.987 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.2 0.107 0.376 0.678 0.879 0.967 0.994 0.999 1.000 1.000 1.000 1.000

0.25 0.056 0.244 0.526 0.776 0.922 0.980 0.996 1.000 1.000 1.000 1.000

0.3 0.028 0.149 0.383 0.650 0.850 0.953 0.989 0.998 1.000 1.000 1.000

0.4 0.006 0.046 0.167 0.382 0.633 0.834 0.945 0.988 0.998 1.000 1.000

θ 0.5 0.001 0.011 0.055 0.172 0.377 0.623 0.828 0.945 0.989 0.999 1.000

0.6 0.000 0.002 0.012 0.055 0.166 0.367 0.618 0.833 0.954 0.994 1.000

0.7 0.000 0.000 0.002 0.011 0.047 0.150 0.350 0.617 0.851 0.972 1.000

0.75 0.000 0.000 0.000 0.004 0.020 0.078 0.224 0.474 0.756 0.944 1.000

0.8 0.000 0.000 0.000 0.001 0.006 0.033 0.121 0.322 0.624 0.893 1.000

0.9 0.000 0.000 0.000 0.000 0.000 0.002 0.013 0.070 0.264 0.651 1.000

0.95 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.012 0.086 0.401 1.000

0.99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.004 0.096 1.000

n = 11

0 1 2 3 4 5 6 7 8 9 10 11

0.895 0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.569 0.898 0.985 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.314 0.697 0.910 0.981 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.086 0.322 0.617 0.839 0.950 0.988 0.998 1.000 1.000 1.000 1.000 1.000

0.042 0.197 0.455 0.713 0.885 0.966 0.992 0.999 1.000 1.000 1.000 1.000

0.020 0.113 0.313 0.570 0.790 0.922 0.978 0.996 0.999 1.000 1.000 1.000

0.004 0.030 0.119 0.296 0.533 0.753 0.901 0.971 0.994 0.999 1.000 1.000

0.000 0.006 0.033 0.113 0.274 0.500 0.726 0.887 0.967 0.994 1.000 1.000

0.000 0.001 0.006 0.029 0.099 0.247 0.467 0.704 0.881 0.970 0.996 1.000

0.000 0.000 0.001 0.004 0.022 0.078 0.210 0.430 0.687 0.887 0.980 1.000

0.000 0.000 0.000 0.001 0.008 0.034 0.115 0.287 0.545 0.803 0.958 1.000

0.000 0.000 0.000 0.000 0.002 0.012 0.050 0.161 0.383 0.678 0.914 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.019 0.090 0.303 0.686 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.015 0.102 0.431 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.105 1.000

n = 12

0 1 2 3 4 5 6 7 8 9 10 11 12

0.886 0.994 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.540 0.882 0.980 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.282 0.659 0.889 0.974 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.069 0.275 0.558 0.795 0.927 0.981 0.996 0.999 1.000 1.000 1.000 1.000 1.000

0.032 0.158 0.391 0.649 0.842 0.946 0.986 0.997 1.000 1.000 1.000 1.000 1.000

0.014 0.085 0.253 0.493 0.724 0.882 0.961 0.991 0.998 1.000 1.000 1.000 1.000

0.002 0.020 0.083 0.225 0.438 0.665 0.842 0.943 0.985 0.997 1.000 1.000 1.000

0.000 0.003 0.019 0.073 0.194 0.387 0.613 0.806 0.927 0.981 0.997 1.000 1.000

0.000 0.000 0.003 0.015 0.057 0.158 0.335 0.562 0.775 0.917 0.980 0.998 1.000

0.000 0.000 0.000 0.002 0.009 0.039 0.118 0.276 0.507 0.747 0.915 0.986 1.000

0.000 0.000 0.000 0.000 0.003 0.014 0.054 0.158 0.351 0.609 0.842 0.968 1.000

0.000 0.000 0.000 0.000 0.001 0.004 0.019 0.073 0.205 0.442 0.725 0.931 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.026 0.111 0.341 0.718 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.020 0.118 0.460 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.114 1.000

n = 13

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.878 0.993 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.513 0.865 0.975 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.254 0.621 0.866 0.966 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.055 0.234 0.502 0.747 0.901 0.970 0.993 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.024 0.127 0.333 0.584 0.794 0.920 0.976 0.994 0.999 1.000 1.000 1.000 1.000 1.000

0.010 0.064 0.202 0.421 0.654 0.835 0.938 0.982 0.996 0.999 1.000 1.000 1.000 1.000

0.001 0.013 0.058 0.169 0.353 0.574 0.771 0.902 0.968 0.992 0.999 1.000 1.000 1.000

0.000 0.002 0.011 0.046 0.133 0.291 0.500 0.709 0.867 0.954 0.989 0.998 1.000 1.000

0.000 0.000 0.001 0.008 0.032 0.098 0.229 0.426 0.647 0.831 0.942 0.987 0.999 1.000

0.000 0.000 0.000 0.001 0.004 0.018 0.062 0.165 0.346 0.579 0.798 0.936 0.990 1.000

0.000 0.000 0.000 0.000 0.001 0.006 0.024 0.080 0.206 0.416 0.667 0.873 0.976 1.000

0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.030 0.099 0.253 0.498 0.766 0.945 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.034 0.134 0.379 0.746 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.025 0.135 0.487 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.122 1.000

n = 14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.869 0.992 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.488 0.847 0.970 0.996 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.229 0.585 0.842 0.956 0.991 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.044 0.198 0.448 0.698 0.870 0.956 0.988 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.018 0.101 0.281 0.521 0.742 0.888 0.962 0.990 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.007 0.047 0.161 0.355 0.584 0.781 0.907 0.969 0.992 0.998 1.000 1.000 1.000 1.000 1.000

0.001 0.008 0.040 0.124 0.279 0.486 0.692 0.850 0.942 0.982 0.996 0.999 1.000 1.000 1.000

0.000 0.001 0.006 0.029 0.090 0.212 0.395 0.605 0.788 0.910 0.971 0.994 0.999 1.000 1.000

0.000 0.000 0.001 0.004 0.018 0.058 0.150 0.308 0.514 0.721 0.876 0.960 0.992 0.999 1.000

0.000 0.000 0.000 0.000 0.002 0.008 0.031 0.093 0.219 0.416 0.645 0.839 0.953 0.993 1.000

0.000 0.000 0.000 0.000 0.000 0.002 0.010 0.038 0.112 0.258 0.479 0.719 0.899 0.982 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.012 0.044 0.130 0.302 0.552 0.802 0.956 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.009 0.044 0.158 0.415 0.771 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.004 0.030 0.153 0.512 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.008 0.131 1.000

n = 10

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

8

Tabla 2C. Probabilidades acumuladas p de la distribuci´ on binomial (n = 15, 16, 17, 18). x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.01 0.860 0.990 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.05 0.463 0.829 0.964 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.1 0.206 0.549 0.816 0.944 0.987 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.2 0.035 0.167 0.398 0.648 0.836 0.939 0.982 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.25 0.013 0.080 0.236 0.461 0.686 0.852 0.943 0.983 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.3 0.005 0.035 0.127 0.297 0.515 0.722 0.869 0.950 0.985 0.996 0.999 1.000 1.000 1.000 1.000 1.000

0.4 0.000 0.005 0.027 0.091 0.217 0.403 0.610 0.787 0.905 0.966 0.991 0.998 1.000 1.000 1.000 1.000

θ 0.5 0.000 0.000 0.004 0.018 0.059 0.151 0.304 0.500 0.696 0.849 0.941 0.982 0.996 1.000 1.000 1.000

0.6 0.000 0.000 0.000 0.002 0.009 0.034 0.095 0.213 0.390 0.597 0.783 0.909 0.973 0.995 1.000 1.000

0.7 0.000 0.000 0.000 0.000 0.001 0.004 0.015 0.050 0.131 0.278 0.485 0.703 0.873 0.965 0.995 1.000

0.75 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.017 0.057 0.148 0.314 0.539 0.764 0.920 0.987 1.000

0.8 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.018 0.061 0.164 0.352 0.602 0.833 0.965 1.000

0.9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.013 0.056 0.184 0.451 0.794 1.000

0.95 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.036 0.171 0.537 1.000

0.99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.010 0.140 1.000

n = 16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.851 0.989 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.440 0.811 0.957 0.993 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.185 0.515 0.789 0.932 0.983 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.028 0.141 0.352 0.598 0.798 0.918 0.973 0.993 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.010 0.063 0.197 0.405 0.630 0.810 0.920 0.973 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.003 0.026 0.099 0.246 0.450 0.660 0.825 0.926 0.974 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.003 0.018 0.065 0.167 0.329 0.527 0.716 0.858 0.942 0.981 0.995 0.999 1.000 1.000 1.000 1.000

0.000 0.000 0.002 0.011 0.038 0.105 0.227 0.402 0.598 0.773 0.895 0.962 0.989 0.998 1.000 1.000 1.000

0.000 0.000 0.000 0.001 0.005 0.019 0.058 0.142 0.284 0.473 0.671 0.833 0.935 0.982 0.997 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.026 0.074 0.175 0.340 0.550 0.754 0.901 0.974 0.997 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.027 0.080 0.190 0.370 0.595 0.803 0.937 0.990 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.027 0.082 0.202 0.402 0.648 0.859 0.972 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.017 0.068 0.211 0.485 0.815 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.043 0.189 0.560 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.011 0.149 1.000

n = 17

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0.843 0.988 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.418 0.792 0.950 0.991 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.167 0.482 0.762 0.917 0.978 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.023 0.118 0.310 0.549 0.758 0.894 0.962 0.989 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.008 0.050 0.164 0.353 0.574 0.765 0.893 0.960 0.988 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.002 0.019 0.077 0.202 0.389 0.597 0.775 0.895 0.960 0.987 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.002 0.012 0.046 0.126 0.264 0.448 0.641 0.801 0.908 0.965 0.989 0.997 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.001 0.006 0.025 0.072 0.166 0.315 0.500 0.685 0.834 0.928 0.975 0.994 0.999 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.003 0.011 0.035 0.092 0.199 0.359 0.552 0.736 0.874 0.954 0.988 0.998 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.013 0.040 0.105 0.225 0.403 0.611 0.798 0.923 0.981 0.998 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.012 0.040 0.107 0.235 0.426 0.647 0.836 0.950 0.992 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.011 0.038 0.106 0.242 0.451 0.690 0.882 0.977 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.022 0.083 0.238 0.518 0.833 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.009 0.050 0.208 0.582 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.012 0.157 1.000

n = 18

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.835 0.986 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.397 0.774 0.942 0.989 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.150 0.450 0.734 0.902 0.972 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.018 0.099 0.271 0.501 0.716 0.867 0.949 0.984 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.006 0.039 0.135 0.306 0.519 0.717 0.861 0.943 0.981 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.002 0.014 0.060 0.165 0.333 0.534 0.722 0.859 0.940 0.979 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.001 0.008 0.033 0.094 0.209 0.374 0.563 0.737 0.865 0.942 0.980 0.994 0.999 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.001 0.004 0.015 0.048 0.119 0.240 0.407 0.593 0.760 0.881 0.952 0.985 0.996 0.999 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.001 0.006 0.020 0.058 0.135 0.263 0.437 0.626 0.791 0.906 0.967 0.992 0.999 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.021 0.060 0.141 0.278 0.466 0.667 0.835 0.940 0.986 0.998 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.019 0.057 0.139 0.283 0.481 0.694 0.865 0.961 0.994 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.016 0.051 0.133 0.284 0.499 0.729 0.901 0.982 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.028 0.098 0.266 0.550 0.850 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.011 0.058 0.226 0.603 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.014 0.165 1.000

n = 15

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

9

Tabla 2D. Probabilidades acumuladas p de la distribuci´ on binomial (n = 19, 20, 21). x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

0.01 0.826 0.985 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.05 0.377 0.755 0.933 0.987 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.1 0.135 0.420 0.705 0.885 0.965 0.991 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.2 0.014 0.083 0.237 0.455 0.673 0.837 0.932 0.977 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.25 0.004 0.031 0.111 0.263 0.465 0.668 0.825 0.923 0.971 0.991 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.3 0.001 0.010 0.046 0.133 0.282 0.474 0.666 0.818 0.916 0.967 0.989 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.4 0.000 0.001 0.005 0.023 0.070 0.163 0.308 0.488 0.667 0.814 0.912 0.965 0.988 0.997 0.999 1.000 1.000 1.000 1.000 1.000

θ 0.5 0.000 0.000 0.000 0.002 0.010 0.032 0.084 0.180 0.324 0.500 0.676 0.820 0.916 0.968 0.990 0.998 1.000 1.000 1.000 1.000

0.6 0.000 0.000 0.000 0.000 0.001 0.003 0.012 0.035 0.088 0.186 0.333 0.512 0.692 0.837 0.930 0.977 0.995 0.999 1.000 1.000

0.7 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.011 0.033 0.084 0.182 0.334 0.526 0.718 0.867 0.954 0.990 0.999 1.000

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.009 0.029 0.077 0.175 0.332 0.535 0.737 0.889 0.969 0.996 1.000

0.8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.023 0.068 0.163 0.327 0.545 0.763 0.917 0.986 1.000

0.9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.009 0.035 0.115 0.295 0.580 0.865 1.000

0.95 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.013 0.067 0.245 0.623 1.000

0.99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.015 0.174 1.000

n = 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.818 0.983 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.358 0.736 0.925 0.984 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.122 0.392 0.677 0.867 0.957 0.989 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.012 0.069 0.206 0.411 0.630 0.804 0.913 0.968 0.990 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.003 0.024 0.091 0.225 0.415 0.617 0.786 0.898 0.959 0.986 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.001 0.008 0.035 0.107 0.238 0.416 0.608 0.772 0.887 0.952 0.983 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.001 0.004 0.016 0.051 0.126 0.250 0.416 0.596 0.755 0.872 0.943 0.979 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.001 0.006 0.021 0.058 0.132 0.252 0.412 0.588 0.748 0.868 0.942 0.979 0.994 0.999 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.021 0.057 0.128 0.245 0.404 0.584 0.750 0.874 0.949 0.984 0.996 0.999 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.017 0.048 0.113 0.228 0.392 0.584 0.762 0.893 0.965 0.992 0.999 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.014 0.041 0.102 0.214 0.383 0.585 0.775 0.909 0.976 0.997 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.010 0.032 0.087 0.196 0.370 0.589 0.794 0.931 0.988 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.011 0.043 0.133 0.323 0.608 0.878 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.016 0.075 0.264 0.642 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.017 0.182 1.000

n = 21

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

0.810 0.981 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.341 0.717 0.915 0.981 0.997 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.109 0.365 0.648 0.848 0.948 0.986 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.009 0.058 0.179 0.370 0.586 0.769 0.891 0.957 0.986 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.002 0.019 0.075 0.192 0.367 0.567 0.744 0.870 0.944 0.979 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.001 0.006 0.027 0.086 0.198 0.363 0.551 0.723 0.852 0.932 0.974 0.991 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.002 0.011 0.037 0.096 0.200 0.350 0.524 0.691 0.826 0.915 0.965 0.988 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.001 0.004 0.013 0.039 0.095 0.192 0.332 0.500 0.668 0.808 0.905 0.961 0.987 0.996 0.999 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.012 0.035 0.085 0.174 0.309 0.476 0.650 0.800 0.904 0.963 0.989 0.998 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.009 0.026 0.068 0.148 0.277 0.449 0.637 0.802 0.914 0.973 0.994 0.999 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.021 0.056 0.130 0.256 0.433 0.633 0.808 0.925 0.981 0.998 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.014 0.043 0.109 0.231 0.414 0.630 0.821 0.942 0.991 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.014 0.052 0.152 0.352 0.635 0.891 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.019 0.085 0.283 0.659 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.019 0.190 1.000

n = 19

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

10

Tabla 2E. Probabilidades acumuladas de la distribuci´ on binomial (n = 22, 23).

n = 22

n = 23

x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

0.01 0.802 0.980 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.05 0.324 0.698 0.905 0.978 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.1 0.098 0.339 0.620 0.828 0.938 0.982 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.2 0.007 0.048 0.154 0.332 0.543 0.733 0.867 0.944 0.980 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.25 0.002 0.015 0.061 0.162 0.323 0.517 0.699 0.838 0.925 0.970 0.990 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.3 0.000 0.004 0.021 0.068 0.165 0.313 0.494 0.671 0.814 0.908 0.961 0.986 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.4 0.000 0.000 0.002 0.008 0.027 0.072 0.158 0.290 0.454 0.624 0.772 0.879 0.945 0.979 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000

θ 0.5 0.000 0.000 0.000 0.000 0.002 0.008 0.026 0.067 0.143 0.262 0.416 0.584 0.738 0.857 0.933 0.974 0.992 0.998 1.000 1.000 1.000 1.000 1.000

0.6 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.021 0.055 0.121 0.228 0.376 0.546 0.710 0.842 0.928 0.973 0.992 0.998 1.000 1.000 1.000

0.7 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.014 0.039 0.092 0.186 0.329 0.506 0.687 0.835 0.932 0.979 0.996 1.000 1.000

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.010 0.030 0.075 0.162 0.301 0.483 0.677 0.838 0.939 0.985 0.998 1.000

0.8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.020 0.056 0.133 0.267 0.457 0.668 0.846 0.952 0.993 1.000

0.9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.018 0.062 0.172 0.380 0.661 0.902 1.000

0.95 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.022 0.095 0.302 0.676 1.000

0.99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.020 0.198 1.000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

0.794 0.978 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.307 0.679 0.895 0.974 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.089 0.315 0.592 0.807 0.927 0.977 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.006 0.040 0.133 0.297 0.501 0.695 0.840 0.928 0.973 0.991 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.001 0.012 0.049 0.137 0.283 0.468 0.654 0.804 0.904 0.959 0.985 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.003 0.016 0.054 0.136 0.269 0.440 0.618 0.771 0.880 0.945 0.979 0.993 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.001 0.005 0.019 0.054 0.124 0.237 0.388 0.556 0.713 0.836 0.919 0.965 0.987 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.001 0.005 0.017 0.047 0.105 0.202 0.339 0.500 0.661 0.798 0.895 0.953 0.983 0.995 0.999 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.013 0.035 0.081 0.164 0.287 0.444 0.612 0.763 0.876 0.946 0.981 0.995 0.999 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.007 0.021 0.055 0.120 0.229 0.382 0.560 0.731 0.864 0.946 0.984 0.997 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.015 0.041 0.096 0.196 0.346 0.532 0.717 0.863 0.951 0.988 0.999 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.009 0.027 0.072 0.160 0.305 0.499 0.703 0.867 0.960 0.994 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.023 0.073 0.193 0.408 0.685 0.911 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.026 0.105 0.321 0.693 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.022 0.206 1.000

c E. Barrios y J. A. Garc´ıa

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Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

11

Tabla 2F. Probabilidades acumuladas de la distribuci´ on binomial (n = 24, 25).

n = 24

n = 25

x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.01 0.786 0.976 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.05 0.292 0.661 0.884 0.970 0.994 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.1 0.080 0.292 0.564 0.786 0.915 0.972 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.2 0.005 0.033 0.115 0.264 0.460 0.656 0.811 0.911 0.964 0.987 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.25 0.001 0.009 0.040 0.115 0.247 0.422 0.607 0.766 0.879 0.945 0.979 0.993 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.3 0.000 0.002 0.012 0.042 0.111 0.229 0.389 0.565 0.725 0.847 0.926 0.969 0.988 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.4 0.000 0.000 0.001 0.004 0.013 0.040 0.096 0.192 0.328 0.489 0.650 0.787 0.886 0.947 0.978 0.992 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000

θ 0.5 0.000 0.000 0.000 0.000 0.001 0.003 0.011 0.032 0.076 0.154 0.271 0.419 0.581 0.729 0.846 0.924 0.968 0.989 0.997 0.999 1.000 1.000 1.000 1.000 1.000

0.6 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.008 0.022 0.053 0.114 0.213 0.350 0.511 0.672 0.808 0.904 0.960 0.987 0.996 0.999 1.000 1.000 1.000

0.7 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.012 0.031 0.074 0.153 0.275 0.435 0.611 0.771 0.889 0.958 0.988 0.998 1.000 1.000

0.75 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.007 0.021 0.055 0.121 0.234 0.393 0.578 0.753 0.885 0.960 0.991 0.999 1.000

0.8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.013 0.036 0.089 0.189 0.344 0.540 0.736 0.885 0.967 0.995 1.000

0.9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.028 0.085 0.214 0.436 0.708 0.920 1.000

0.95 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.030 0.116 0.339 0.708 1.000

0.99 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.024 0.214 1.000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 24

0.778 0.974 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.277 0.642 0.873 0.966 0.993 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.072 0.271 0.537 0.764 0.902 0.967 0.991 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.004 0.027 0.098 0.234 0.421 0.617 0.780 0.891 0.953 0.983 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.001 0.007 0.032 0.096 0.214 0.378 0.561 0.727 0.851 0.929 0.970 0.989 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.002 0.009 0.033 0.090 0.193 0.341 0.512 0.677 0.811 0.902 0.956 0.983 0.994 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.002 0.009 0.029 0.074 0.154 0.274 0.425 0.586 0.732 0.846 0.922 0.966 0.987 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.002 0.007 0.022 0.054 0.115 0.212 0.345 0.500 0.655 0.788 0.885 0.946 0.978 0.993 0.998 1.000 1.000 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.013 0.034 0.078 0.154 0.268 0.414 0.575 0.726 0.846 0.926 0.971 0.991 0.998 1.000 1.000 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.017 0.044 0.098 0.189 0.323 0.488 0.659 0.807 0.910 0.967 0.991 0.998 1.000 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.011 0.030 0.071 0.149 0.273 0.439 0.622 0.786 0.904 0.968 0.993 0.999 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.006 0.017 0.047 0.109 0.220 0.383 0.579 0.766 0.902 0.973 0.996 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.009 0.033 0.098 0.236 0.463 0.729 0.928 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.007 0.034 0.127 0.358 0.723 1.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.026 0.222 1.000

c E. Barrios y J. A. Garc´ıa

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.2.

12

Distribuci´ on Poisson

X ∼ Poisson(λ) p = P (X ≤ x) =

x X λk e−λ

k=0

k!

α

p

=1−α

0

x

Tabla 2A. Probabilidades acumuladas p de la distribuci´ on Poisson. λ x 0 1 2 3 4 5 6

0.1 0.905 0.995 1.000 1.000 1.000 1.000 1.000

0.2 0.819 0.982 0.999 1.000 1.000 1.000 1.000

0.3 0.741 0.963 0.996 1.000 1.000 1.000 1.000

0.4 0.670 0.938 0.992 0.999 1.000 1.000 1.000

0.5 0.607 0.910 0.986 0.998 1.000 1.000 1.000

0.6 0.549 0.878 0.977 0.997 1.000 1.000 1.000

0.7 0.497 0.844 0.966 0.994 0.999 1.000 1.000

0.8 0.449 0.809 0.953 0.991 0.999 1.000 1.000

0.9 0.407 0.772 0.937 0.987 0.998 1.000 1.000

1.0 0.368 0.736 0.920 0.981 0.996 0.999 1.000

Tabla 2B. Probabilidades acumuladas p de la distribuci´ on Poisson. λ x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

2 0.135 0.406 0.677 0.857 0.947 0.983 0.995 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

c E. Barrios y J. A. Garc´ıa

3 0.050 0.199 0.423 0.647 0.815 0.916 0.966 0.988 0.996 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

4 0.018 0.092 0.238 0.433 0.629 0.785 0.889 0.949 0.979 0.992 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

5 0.007 0.040 0.125 0.265 0.440 0.616 0.762 0.867 0.932 0.968 0.986 0.995 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

6 0.002 0.017 0.062 0.151 0.285 0.446 0.606 0.744 0.847 0.916 0.957 0.980 0.991 0.996 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

7 0.001 0.007 0.030 0.082 0.173 0.301 0.450 0.599 0.729 0.830 0.901 0.947 0.973 0.987 0.994 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

8 0.000 0.003 0.014 0.042 0.100 0.191 0.313 0.453 0.593 0.717 0.816 0.888 0.936 0.966 0.983 0.992 0.996 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

9 0.000 0.001 0.006 0.021 0.055 0.116 0.207 0.324 0.456 0.587 0.706 0.803 0.876 0.926 0.959 0.978 0.989 0.995 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

10 0.000 0.000 0.003 0.010 0.029 0.067 0.130 0.220 0.333 0.458 0.583 0.697 0.792 0.864 0.917 0.951 0.973 0.986 0.993 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

15 0.000 0.000 0.000 0.000 0.001 0.003 0.008 0.018 0.037 0.070 0.118 0.185 0.268 0.363 0.466 0.568 0.664 0.749 0.819 0.875 0.917 0.947 0.967 0.981 0.989 0.994 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.005 0.011 0.021 0.039 0.066 0.105 0.157 0.221 0.297 0.381 0.470 0.559 0.644 0.721 0.787 0.843 0.888 0.922 0.948 0.966 0.978 0.987 0.992 0.995 0.997 0.999 0.999 1.000

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.3.

13

Distribuci´ on Normal Est´ andar Z ∼ N(0, 1) p = P (Z ≤ z) =

donde

Z

z

−∞

φZ (u)du = 1 − α

α

p

1 2 1 φZ (u) = √ e− 2 u 2π

0 z

Nota: Si X ∼ N(µ, σ 2 ), entonces Z = (X − µ)/σ ∼ N(0, 1). Luego, x−µ P (X ≤ x) = P Z ≤ σ

Tabla 2A. Probabilidades acumuladas p de la distribuci´ on normal est´ andar. z −3.4 −3.3 −3.2 −3.1 −3.0 −2.9 −2.8 −2.7 −2.6 −2.5 −2.4 −2.3 −2.2 −2.1 −2.0 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 −0.0

0.09 0.0002 0.0003 0.0005 0.0007 0.0010 0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183 0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379 0.1611 0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641

c E. Barrios y J. A. Garc´ıa

0.08 0.0003 0.0004 0.0005 0.0007 0.0010 0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188 0.0239 0.0301 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401 0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681

0.07 0.0003 0.0004 0.0005 0.0008 0.0011 0.0015 0.0021 0.0028 0.0038 0.0051 0.0068 0.0089 0.0116 0.0150 0.0192 0.0244 0.0307 0.0384 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423 0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721

0.06 0.0003 0.0004 0.0006 0.0008 0.0011 0.0015 0.0021 0.0029 0.0039 0.0052 0.0069 0.0091 0.0119 0.0154 0.0197 0.0250 0.0314 0.0392 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.1446 0.1685 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761

0.05 0.0003 0.0004 0.0006 0.0008 0.0011 0.0016 0.0022 0.0030 0.0040 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202 0.0256 0.0322 0.0401 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 0.1469 0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801

0.04 0.0003 0.0004 0.0006 0.0008 0.0012 0.0016 0.0023 0.0031 0.0041 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207 0.0262 0.0329 0.0409 0.0505 0.0618 0.0749 0.0901 0.1075 0.1271 0.1492 0.1736 0.2005 0.2296 0.2611 0.2946 0.3300 0.3669 0.4052 0.4443 0.4840

0.03 0.0003 0.0004 0.0006 0.0009 0.0012 0.0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0099 0.0129 0.0166 0.0212 0.0268 0.0336 0.0418 0.0516 0.0630 0.0764 0.0918 0.1093 0.1292 0.1515 0.1762 0.2033 0.2327 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880

0.02 0.0003 0.0005 0.0006 0.0009 0.0013 0.0018 0.0024 0.0033 0.0044 0.0059 0.0078 0.0102 0.0132 0.0170 0.0217 0.0274 0.0344 0.0427 0.0526 0.0643 0.0778 0.0934 0.1112 0.1314 0.1539 0.1788 0.2061 0.2358 0.2676 0.3015 0.3372 0.3745 0.4129 0.4522 0.4920

0.01 0.0003 0.0005 0.0007 0.0009 0.0013 0.0018 0.0025 0.0034 0.0045 0.0060 0.0080 0.0104 0.0136 0.0174 0.0222 0.0281 0.0351 0.0436 0.0537 0.0655 0.0793 0.0951 0.1131 0.1335 0.1562 0.1814 0.2090 0.2389 0.2709 0.3050 0.3409 0.3783 0.4168 0.4562 0.4960

0.00 0.0003 0.0005 0.0007 0.0010 0.0013 0.0019 0.0026 0.0035 0.0047 0.0062 0.0082 0.0107 0.0139 0.0179 0.0228 0.0287 0.0359 0.0446 0.0548 0.0668 0.0808 0.0968 0.1151 0.1357 0.1587 0.1841 0.2119 0.2420 0.2743 0.3085 0.3446 0.3821 0.4207 0.4602 0.5000

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

14

Tabla 2B. Probabilidades acumuladas p de la distribuci´ on normal est´ andar. z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4

0.00 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.9990 0.9993 0.9995 0.9997

c E. Barrios y J. A. Garc´ıa

0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.9991 0.9993 0.9995 0.9997

0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987 0.9991 0.9994 0.9995 0.9997

0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.9991 0.9994 0.9996 0.9997

0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997

0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.9992 0.9994 0.9996 0.9997

0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.9992 0.9994 0.9996 0.9997

0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997

0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.9997

0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.4.

15

Distribuci´ on Gamma X ∼ Γ(κ, 1) p = P (X ≤ x) =

donde

Z

0

x

φΓ (u)du = 1 − α

α

p

1 uκ−1 e−u φΓ (u) = Γ(κ)

0

x

Nota: Si Y ∼ Γ(κ, λ), entonces, X = λY ∼ Γ(κ, 1). Luego, P (Y ≤ y) = P (X ≤ λy)

Tabla 2. Probabilidades acumuladas p de la distribuci´ on κ x 0.1 0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

1 0.095 0.393 0.632 0.865 0.950 0.982 0.993 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

c E. Barrios y J. A. Garc´ıa

2 0.005 0.090 0.264 0.594 0.801 0.908 0.960 0.983 0.993 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

3 0.000 0.014 0.080 0.323 0.577 0.762 0.875 0.938 0.970 0.986 0.994 0.997 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

4 0.000 0.002 0.019 0.143 0.353 0.567 0.735 0.849 0.918 0.958 0.979 0.990 0.995 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

5 0.000 0.000 0.004 0.053 0.185 0.371 0.560 0.715 0.827 0.900 0.945 0.971 0.985 0.992 0.996 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

6 0.000 0.000 0.001 0.017 0.084 0.215 0.384 0.554 0.699 0.809 0.884 0.933 0.962 0.980 0.989 0.994 0.997 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

7 0.000 0.000 0.000 0.005 0.034 0.111 0.238 0.394 0.550 0.687 0.793 0.870 0.921 0.954 0.974 0.986 0.992 0.996 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

8 0.000 0.000 0.000 0.001 0.012 0.051 0.133 0.256 0.401 0.547 0.676 0.780 0.857 0.910 0.946 0.968 0.982 0.990 0.995 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

9 0.000 0.000 0.000 0.000 0.004 0.021 0.068 0.153 0.271 0.407 0.544 0.667 0.768 0.845 0.900 0.938 0.963 0.978 0.987 0.993 0.996 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

10 0.000 0.000 0.000 0.000 0.001 0.008 0.032 0.084 0.170 0.283 0.413 0.542 0.659 0.758 0.834 0.891 0.930 0.957 0.974 0.985 0.991 0.995 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

15 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.006 0.017 0.041 0.083 0.146 0.228 0.325 0.430 0.534 0.632 0.719 0.792 0.850 0.895 0.928 0.952 0.969 0.980 0.988 0.992 0.995 0.997 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.009 0.021 0.043 0.077 0.125 0.188 0.264 0.349 0.439 0.530 0.616 0.694 0.762 0.820 0.866 0.903 0.931 0.952 0.967 0.978 0.986 0.991 0.994 0.996 0.998 0.999 0.999 0.999 1.000

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.5.

16

Distribuci´ on t de Student T ∼ tν

siendo ν los grados de libertad. Z t φT (u)du = 1 − α p = P (T ≤ t) = −∞

donde

α

p

− ν+1 2 u2 1 Γ ν+1 2 1+ φT (u) = √ ν ν νπ Γ 2

0

Tabla 2. Valores cr´ıticos t(α;ν) de la distribuci´ on t de Student.

ν 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 75 100 125 ∞

0.75

0.80

0.90

0.95

0.975

0.25 1.000 0.816 0.765 0.741 0.727 0.718 0.711 0.706 0.703 0.700 0.697 0.695 0.694 0.692 0.691 0.690 0.689 0.688 0.688 0.687 0.686 0.686 0.685 0.685 0.684 0.684 0.684 0.683 0.683 0.683 0.681 0.679 0.678 0.677 0.676 0.674

0.20 1.376 1.061 0.978 0.941 0.920 0.906 0.896 0.889 0.883 0.879 0.876 0.873 0.870 0.868 0.866 0.865 0.863 0.862 0.861 0.860 0.859 0.858 0.858 0.857 0.856 0.856 0.855 0.855 0.854 0.854 0.851 0.849 0.846 0.845 0.845 0.842

0.10 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.303 1.299 1.293 1.290 1.288 1.282

0.05 6.314 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.684 1.676 1.665 1.660 1.657 1.645

0.025 12.706 4.303 3.182 2.776 2.571 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.021 2.009 1.992 1.984 1.979 1.960

c E. Barrios y J. A. Garc´ıa

p 0.99 α 0.01 31.821 6.965 4.541 3.747 3.365 3.143 2.998 2.896 2.821 2.764 2.718 2.681 2.650 2.624 2.602 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.423 2.403 2.377 2.364 2.357 2.326

0.995

0.999

0.9995

0.9999

0.005 63.657 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.704 2.678 2.643 2.626 2.616 2.576

0.001 318.309 22.327 10.215 7.173 5.893 5.208 4.785 4.501 4.297 4.144 4.025 3.930 3.852 3.787 3.733 3.686 3.646 3.610 3.579 3.552 3.527 3.505 3.485 3.467 3.450 3.435 3.421 3.408 3.396 3.385 3.307 3.261 3.202 3.174 3.157 3.090

0.0005 636.619 31.599 12.924 8.610 6.869 5.959 5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.496 3.425 3.390 3.370 3.291

0.0001 3183.099 70.700 22.204 13.034 9.678 8.025 7.063 6.442 6.010 5.694 5.453 5.263 5.111 4.985 4.880 4.791 4.714 4.648 4.590 4.539 4.493 4.452 4.415 4.382 4.352 4.324 4.299 4.275 4.254 4.234 4.094 4.014 3.911 3.862 3.832 3.719

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.6.

17

Distribuci´ on χ2 Ji-Cuadrada Y ∼ χ2ν

siendo ν los grados de libertad. Z y φχ2 (u)du = 1 − α p = P (Y ≤ y) = 0

α

p

donde φχ2 (u) =

1 uν/2−1 e−u/2 2ν/2 Γ (ν/2)

0

x

Tabla 2. Valores cr´ıticos χ2(α;ν) de la distribuci´ on χ2ν Ji-Cuadrada. p 0.005

0.01

0.025

0.05

0.1

0.90

0.95

0.975

0.99

0.995

0.10 2.706 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.087 40.256 41.422 42.585 43.745 44.903 46.059 47.212 48.363 49.513 50.660 51.805 63.167 91.061 118.498

0.05 3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 44.985 46.194 47.400 48.602 49.802 50.998 52.192 53.384 54.572 55.758 67.505 96.217 124.342

0.025 5.024 7.378 9.348 11.143 12.833 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.195 44.461 45.722 46.979 48.232 49.480 50.725 51.966 53.203 54.437 55.668 56.896 58.120 59.342 71.420 100.839 129.561

0.01 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 52.191 53.486 54.776 56.061 57.342 58.619 59.893 61.162 62.428 63.691 76.154 106.393 135.807

0.005 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.559 46.928 48.290 49.645 50.993 52.336 53.672 55.003 56.328 57.648 58.964 60.275 61.581 62.883 64.181 65.476 66.766 79.490 110.286 140.169

α ν 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 50 75 100

0.995 0.000 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787 14.458 15.134 15.815 16.501 17.192 17.887 18.586 19.289 19.996 20.707 27.991 47.206 67.328

c E. Barrios y J. A. Garc´ıa

0.99 0.000 0.020 0.115 0.297 0.554 0.872 1.239 1.646 2.088 2.558 3.053 3.571 4.107 4.660 5.229 5.812 6.408 7.015 7.633 8.260 8.897 9.542 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953 15.655 16.362 17.074 17.789 18.509 19.233 19.960 20.691 21.426 22.164 29.707 49.475 70.065

0.975 0.001 0.051 0.216 0.484 0.831 1.237 1.690 2.180 2.700 3.247 3.816 4.404 5.009 5.629 6.262 6.908 7.564 8.231 8.907 9.591 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 17.539 18.291 19.047 19.806 20.569 21.336 22.106 22.878 23.654 24.433 32.357 52.942 74.222

0.95 0.004 0.103 0.352 0.711 1.145 1.635 2.167 2.733 3.325 3.940 4.575 5.226 5.892 6.571 7.261 7.962 8.672 9.390 10.117 10.851 11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493 19.281 20.072 20.867 21.664 22.465 23.269 24.075 24.884 25.695 26.509 34.764 56.054 77.929

0.90 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.578 6.304 7.042 7.790 8.547 9.312 10.085 10.865 11.651 12.443 13.240 14.041 14.848 15.659 16.473 17.292 18.114 18.939 19.768 20.599 21.434 22.271 23.110 23.952 24.797 25.643 26.492 27.343 28.196 29.051 37.689 59.795 82.358

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.7.

18

Distribuci´ on F F ∼ Fν1 ,ν2

siendo ν1 y ν2 los grados de libertad. p = P (F ≤ f ) =

Z

0

f

φF (u)du = 1 − α

donde φF (u) =

α

p Γ

ν1 +ν2 2

ν /2 ν /2 ν 1 1 ν2 2

Γ(ν1 /2)Γ(ν2 /2)

0

u(ν1 /2)−1 (ν2 + ν1 u)−(ν1 +ν2 )/2

f

Nota: Si F ∼ Fν1 ,ν2 , entonces, p = P (F ≤ F(1−α;ν1 ,ν2 ) ) = P

c E. Barrios y J. A. Garc´ıa

F ≤

1 F(α;ν2 ,ν1 )

=1−α

v.1.00

p = 0.90

α = 0.10 ν1

ν2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 80 100 120 ∞

1 39.86 8.53 5.54 4.54 4.06 3.78 3.59 3.46 3.36 3.29 3.23 3.18 3.14 3.10 3.07 3.05 3.03 3.01 2.99 2.97 2.96 2.95 2.94 2.93 2.92 2.91 2.90 2.89 2.89 2.88 2.84 2.79 2.77 2.76 2.75 2.71

2 49.50 9.00 5.46 4.32 3.78 3.46 3.26 3.11 3.01 2.92 2.86 2.81 2.76 2.73 2.70 2.67 2.64 2.62 2.61 2.59 2.57 2.56 2.55 2.54 2.53 2.52 2.51 2.50 2.50 2.49 2.44 2.39 2.37 2.36 2.35 2.30

3 53.59 9.16 5.39 4.19 3.62 3.29 3.07 2.92 2.81 2.73 2.66 2.61 2.56 2.52 2.49 2.46 2.44 2.42 2.40 2.38 2.36 2.35 2.34 2.33 2.32 2.31 2.30 2.29 2.28 2.28 2.23 2.18 2.15 2.14 2.13 2.08

4 55.83 9.24 5.34 4.11 3.52 3.18 2.96 2.81 2.69 2.61 2.54 2.48 2.43 2.39 2.36 2.33 2.31 2.29 2.27 2.25 2.23 2.22 2.21 2.19 2.18 2.17 2.17 2.16 2.15 2.14 2.09 2.04 2.02 2.00 1.99 1.95

5 57.24 9.29 5.31 4.05 3.45 3.11 2.88 2.73 2.61 2.52 2.45 2.39 2.35 2.31 2.27 2.24 2.22 2.20 2.18 2.16 2.14 2.13 2.11 2.10 2.09 2.08 2.07 2.06 2.06 2.05 2.00 1.95 1.92 1.91 1.90 1.85

6 58.20 9.33 5.28 4.01 3.40 3.05 2.83 2.67 2.55 2.46 2.39 2.33 2.28 2.24 2.21 2.18 2.15 2.13 2.11 2.09 2.08 2.06 2.05 2.04 2.02 2.01 2.00 2.00 1.99 1.98 1.93 1.87 1.85 1.83 1.82 1.77

7 58.91 9.35 5.27 3.98 3.37 3.01 2.78 2.62 2.51 2.41 2.34 2.28 2.23 2.19 2.16 2.13 2.10 2.08 2.06 2.04 2.02 2.01 1.99 1.98 1.97 1.96 1.95 1.94 1.93 1.93 1.87 1.82 1.79 1.78 1.77 1.72

8 59.44 9.37 5.25 3.95 3.34 2.98 2.75 2.59 2.47 2.38 2.30 2.24 2.20 2.15 2.12 2.09 2.06 2.04 2.02 2.00 1.98 1.97 1.95 1.94 1.93 1.92 1.91 1.90 1.89 1.88 1.83 1.77 1.75 1.73 1.72 1.67

9 59.86 9.38 5.24 3.94 3.32 2.96 2.72 2.56 2.44 2.35 2.27 2.21 2.16 2.12 2.09 2.06 2.03 2.00 1.98 1.96 1.95 1.93 1.92 1.91 1.89 1.88 1.87 1.87 1.86 1.85 1.79 1.74 1.71 1.69 1.68 1.63

10 60.19 9.39 5.23 3.92 3.30 2.94 2.70 2.54 2.42 2.32 2.25 2.19 2.14 2.10 2.06 2.03 2.00 1.98 1.96 1.94 1.92 1.90 1.89 1.88 1.87 1.86 1.85 1.84 1.83 1.82 1.76 1.71 1.68 1.66 1.65 1.60

12 60.71 9.41 5.22 3.90 3.27 2.90 2.67 2.50 2.38 2.28 2.21 2.15 2.10 2.05 2.02 1.99 1.96 1.93 1.91 1.89 1.87 1.86 1.84 1.83 1.82 1.81 1.80 1.79 1.78 1.77 1.71 1.66 1.63 1.61 1.60 1.55

15 61.22 9.42 5.20 3.87 3.24 2.87 2.63 2.46 2.34 2.24 2.17 2.10 2.05 2.01 1.97 1.94 1.91 1.89 1.86 1.84 1.83 1.81 1.80 1.78 1.77 1.76 1.75 1.74 1.73 1.72 1.66 1.60 1.57 1.56 1.55 1.49

20 61.74 9.44 5.18 3.84 3.21 2.84 2.59 2.42 2.30 2.20 2.12 2.06 2.01 1.96 1.92 1.89 1.86 1.84 1.81 1.79 1.78 1.76 1.74 1.73 1.72 1.71 1.70 1.69 1.68 1.67 1.61 1.54 1.51 1.49 1.48 1.42

25 62.05 9.45 5.17 3.83 3.19 2.81 2.57 2.40 2.27 2.17 2.10 2.03 1.98 1.93 1.89 1.86 1.83 1.80 1.78 1.76 1.74 1.73 1.71 1.70 1.68 1.67 1.66 1.65 1.64 1.63 1.57 1.50 1.47 1.45 1.44 1.38

50 62.69 9.47 5.15 3.80 3.15 2.77 2.52 2.35 2.22 2.12 2.04 1.97 1.92 1.87 1.83 1.79 1.76 1.74 1.71 1.69 1.67 1.65 1.64 1.62 1.61 1.59 1.58 1.57 1.56 1.55 1.48 1.41 1.38 1.35 1.34 1.26

75 62.90 9.48 5.15 3.78 3.13 2.75 2.51 2.33 2.20 2.10 2.02 1.95 1.89 1.85 1.80 1.77 1.74 1.71 1.69 1.66 1.64 1.63 1.61 1.59 1.58 1.57 1.55 1.54 1.53 1.52 1.45 1.38 1.34 1.32 1.30 1.22

100 63.01 9.48 5.14 3.78 3.13 2.75 2.50 2.32 2.19 2.09 2.01 1.94 1.88 1.83 1.79 1.76 1.73 1.70 1.67 1.65 1.63 1.61 1.59 1.58 1.56 1.55 1.54 1.53 1.52 1.51 1.43 1.36 1.32 1.29 1.28 1.19

∞ 63.32 9.49 5.13 3.76 3.11 2.72 2.47 2.29 2.16 2.06 1.97 1.90 1.85 1.80 1.76 1.72 1.69 1.66 1.63 1.61 1.59 1.57 1.55 1.53 1.52 1.50 1.49 1.48 1.47 1.46 1.38 1.29 1.25 1.22 1.19 1.03

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

Tabla 2A. Valores cr´ıticos F(α;ν1 ,ν2 ) de la distribuci´ on F .

19

v.1.00

p = 0.95

α = 0.05 ν1

ν2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 80 100 120 ∞

1 161.45 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.67 4.60 4.54 4.49 4.45 4.41 4.38 4.35 4.32 4.30 4.28 4.26 4.24 4.23 4.21 4.20 4.18 4.17 4.08 4.00 3.96 3.94 3.92 3.84

2 199.50 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.81 3.74 3.68 3.63 3.59 3.55 3.52 3.49 3.47 3.44 3.42 3.40 3.39 3.37 3.35 3.34 3.33 3.32 3.23 3.15 3.11 3.09 3.07 3.00

3 215.71 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.41 3.34 3.29 3.24 3.20 3.16 3.13 3.10 3.07 3.05 3.03 3.01 2.99 2.98 2.96 2.95 2.93 2.92 2.84 2.76 2.72 2.70 2.68 2.61

4 224.58 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.18 3.11 3.06 3.01 2.96 2.93 2.90 2.87 2.84 2.82 2.80 2.78 2.76 2.74 2.73 2.71 2.70 2.69 2.61 2.53 2.49 2.46 2.45 2.37

5 230.16 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 3.03 2.96 2.90 2.85 2.81 2.77 2.74 2.71 2.68 2.66 2.64 2.62 2.60 2.59 2.57 2.56 2.55 2.53 2.45 2.37 2.33 2.31 2.29 2.21

6 233.99 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.92 2.85 2.79 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.53 2.51 2.49 2.47 2.46 2.45 2.43 2.42 2.34 2.25 2.21 2.19 2.18 2.10

7 236.77 19.35 8.89 6.09 4.88 4.21 3.79 3.50 3.29 3.14 3.01 2.91 2.83 2.76 2.71 2.66 2.61 2.58 2.54 2.51 2.49 2.46 2.44 2.42 2.40 2.39 2.37 2.36 2.35 2.33 2.25 2.17 2.13 2.10 2.09 2.01

8 238.88 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.77 2.70 2.64 2.59 2.55 2.51 2.48 2.45 2.42 2.40 2.37 2.36 2.34 2.32 2.31 2.29 2.28 2.27 2.18 2.10 2.06 2.03 2.02 1.94

9 240.54 19.38 8.81 6.00 4.77 4.10 3.68 3.39 3.18 3.02 2.90 2.80 2.71 2.65 2.59 2.54 2.49 2.46 2.42 2.39 2.37 2.34 2.32 2.30 2.28 2.27 2.25 2.24 2.22 2.21 2.12 2.04 2.00 1.97 1.96 1.88

10 241.88 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.67 2.60 2.54 2.49 2.45 2.41 2.38 2.35 2.32 2.30 2.27 2.25 2.24 2.22 2.20 2.19 2.18 2.16 2.08 1.99 1.95 1.93 1.91 1.83

12 243.91 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.60 2.53 2.48 2.42 2.38 2.34 2.31 2.28 2.25 2.23 2.20 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.00 1.92 1.88 1.85 1.83 1.75

15 245.95 19.43 8.70 5.86 4.62 3.94 3.51 3.22 3.01 2.85 2.72 2.62 2.53 2.46 2.40 2.35 2.31 2.27 2.23 2.20 2.18 2.15 2.13 2.11 2.09 2.07 2.06 2.04 2.03 2.01 1.92 1.84 1.79 1.77 1.75 1.67

20 248.01 19.45 8.66 5.80 4.56 3.87 3.44 3.15 2.94 2.77 2.65 2.54 2.46 2.39 2.33 2.28 2.23 2.19 2.16 2.12 2.10 2.07 2.05 2.03 2.01 1.99 1.97 1.96 1.94 1.93 1.84 1.75 1.70 1.68 1.66 1.57

25 249.26 19.46 8.63 5.77 4.52 3.83 3.40 3.11 2.89 2.73 2.60 2.50 2.41 2.34 2.28 2.23 2.18 2.14 2.11 2.07 2.05 2.02 2.00 1.97 1.96 1.94 1.92 1.91 1.89 1.88 1.78 1.69 1.64 1.62 1.60 1.51

50 251.77 19.48 8.58 5.70 4.44 3.75 3.32 3.02 2.80 2.64 2.51 2.40 2.31 2.24 2.18 2.12 2.08 2.04 2.00 1.97 1.94 1.91 1.88 1.86 1.84 1.82 1.81 1.79 1.77 1.76 1.66 1.56 1.51 1.48 1.46 1.35

75 252.62 19.48 8.56 5.68 4.42 3.73 3.29 2.99 2.77 2.60 2.47 2.37 2.28 2.21 2.14 2.09 2.04 2.00 1.96 1.93 1.90 1.87 1.84 1.82 1.80 1.78 1.76 1.75 1.73 1.72 1.61 1.51 1.45 1.42 1.40 1.28

100 253.04 19.49 8.55 5.66 4.41 3.71 3.27 2.97 2.76 2.59 2.46 2.35 2.26 2.19 2.12 2.07 2.02 1.98 1.94 1.91 1.88 1.85 1.82 1.80 1.78 1.76 1.74 1.73 1.71 1.70 1.59 1.48 1.43 1.39 1.37 1.25

∞ 254.30 19.50 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.41 2.30 2.21 2.13 2.07 2.01 1.96 1.92 1.88 1.84 1.81 1.78 1.76 1.73 1.71 1.69 1.67 1.65 1.64 1.62 1.51 1.39 1.33 1.28 1.26 1.03

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

Tabla 2B. Valores cr´ıticos F(α;ν1 ,ν2 ) de la distribuci´ on F .

20

v.1.00

p = 0.975

α = 0.025 ν1

ν2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 80 100 120 ∞

1 648. 38.51 17.44 12.22 10.01 8.81 8.07 7.57 7.21 6.94 6.72 6.55 6.41 6.30 6.20 6.12 6.04 5.98 5.92 5.87 5.83 5.79 5.75 5.72 5.69 5.66 5.63 5.61 5.59 5.57 5.42 5.29 5.22 5.18 5.15 5.03

2 799. 39.00 16.04 10.65 8.43 7.26 6.54 6.06 5.71 5.46 5.26 5.10 4.97 4.86 4.77 4.69 4.62 4.56 4.51 4.46 4.42 4.38 4.35 4.32 4.29 4.27 4.24 4.22 4.20 4.18 4.05 3.93 3.86 3.83 3.80 3.69

3 864. 39.17 15.44 9.98 7.76 6.60 5.89 5.42 5.08 4.83 4.63 4.47 4.35 4.24 4.15 4.08 4.01 3.95 3.90 3.86 3.82 3.78 3.75 3.72 3.69 3.67 3.65 3.63 3.61 3.59 3.46 3.34 3.28 3.25 3.23 3.12

4 900. 39.25 15.10 9.60 7.39 6.23 5.52 5.05 4.72 4.47 4.28 4.12 4.00 3.89 3.80 3.73 3.66 3.61 3.56 3.51 3.48 3.44 3.41 3.38 3.35 3.33 3.31 3.29 3.27 3.25 3.13 3.01 2.95 2.92 2.89 2.79

5 922. 39.30 14.88 9.36 7.15 5.99 5.29 4.82 4.48 4.24 4.04 3.89 3.77 3.66 3.58 3.50 3.44 3.38 3.33 3.29 3.25 3.22 3.18 3.15 3.13 3.10 3.08 3.06 3.04 3.03 2.90 2.79 2.73 2.70 2.67 2.57

6 937. 39.33 14.73 9.20 6.98 5.82 5.12 4.65 4.32 4.07 3.88 3.73 3.60 3.50 3.41 3.34 3.28 3.22 3.17 3.13 3.09 3.05 3.02 2.99 2.97 2.94 2.92 2.90 2.88 2.87 2.74 2.63 2.57 2.54 2.52 2.41

7 948. 39.36 14.62 9.07 6.85 5.70 4.99 4.53 4.20 3.95 3.76 3.61 3.48 3.38 3.29 3.22 3.16 3.10 3.05 3.01 2.97 2.93 2.90 2.87 2.85 2.82 2.80 2.78 2.76 2.75 2.62 2.51 2.45 2.42 2.39 2.29

8 957. 39.37 14.54 8.98 6.76 5.60 4.90 4.43 4.10 3.85 3.66 3.51 3.39 3.29 3.20 3.12 3.06 3.01 2.96 2.91 2.87 2.84 2.81 2.78 2.75 2.73 2.71 2.69 2.67 2.65 2.53 2.41 2.35 2.32 2.30 2.19

9 963. 39.39 14.47 8.90 6.68 5.52 4.82 4.36 4.03 3.78 3.59 3.44 3.31 3.21 3.12 3.05 2.98 2.93 2.88 2.84 2.80 2.76 2.73 2.70 2.68 2.65 2.63 2.61 2.59 2.57 2.45 2.33 2.28 2.24 2.22 2.11

10 969. 39.40 14.42 8.84 6.62 5.46 4.76 4.30 3.96 3.72 3.53 3.37 3.25 3.15 3.06 2.99 2.92 2.87 2.82 2.77 2.73 2.70 2.67 2.64 2.61 2.59 2.57 2.55 2.53 2.51 2.39 2.27 2.21 2.18 2.16 2.05

12 977. 39.41 14.34 8.75 6.52 5.37 4.67 4.20 3.87 3.62 3.43 3.28 3.15 3.05 2.96 2.89 2.82 2.77 2.72 2.68 2.64 2.60 2.57 2.54 2.51 2.49 2.47 2.45 2.43 2.41 2.29 2.17 2.11 2.08 2.05 1.95

15 985. 39.43 14.25 8.66 6.43 5.27 4.57 4.10 3.77 3.52 3.33 3.18 3.05 2.95 2.86 2.79 2.72 2.67 2.62 2.57 2.53 2.50 2.47 2.44 2.41 2.39 2.36 2.34 2.32 2.31 2.18 2.06 2.00 1.97 1.94 1.83

20 993. 39.45 14.17 8.56 6.33 5.17 4.47 4.00 3.67 3.42 3.23 3.07 2.95 2.84 2.76 2.68 2.62 2.56 2.51 2.46 2.42 2.39 2.36 2.33 2.30 2.28 2.25 2.23 2.21 2.20 2.07 1.94 1.88 1.85 1.82 1.71

25 998. 39.46 14.12 8.50 6.27 5.11 4.40 3.94 3.60 3.35 3.16 3.01 2.88 2.78 2.69 2.61 2.55 2.49 2.44 2.40 2.36 2.32 2.29 2.26 2.23 2.21 2.18 2.16 2.14 2.12 1.99 1.87 1.81 1.77 1.75 1.63

50 1008. 39.48 14.01 8.38 6.14 4.98 4.28 3.81 3.47 3.22 3.03 2.87 2.74 2.64 2.55 2.47 2.41 2.35 2.30 2.25 2.21 2.17 2.14 2.11 2.08 2.05 2.03 2.01 1.99 1.97 1.83 1.70 1.63 1.59 1.56 1.43

75 1011. 39.48 13.97 8.34 6.10 4.94 4.23 3.76 3.43 3.18 2.98 2.82 2.70 2.59 2.50 2.42 2.35 2.30 2.24 2.20 2.16 2.12 2.08 2.05 2.02 2.00 1.97 1.95 1.93 1.91 1.77 1.63 1.56 1.52 1.49 1.35

100 1013. 39.49 13.96 8.32 6.08 4.92 4.21 3.74 3.40 3.15 2.96 2.80 2.67 2.56 2.47 2.40 2.33 2.27 2.22 2.17 2.13 2.09 2.06 2.02 2.00 1.97 1.94 1.92 1.90 1.88 1.74 1.60 1.53 1.48 1.45 1.30

∞ 1018. 39.50 13.90 8.26 6.02 4.85 4.14 3.67 3.33 3.08 2.88 2.73 2.60 2.49 2.40 2.32 2.25 2.19 2.13 2.09 2.04 2.00 1.97 1.94 1.91 1.88 1.85 1.83 1.81 1.79 1.64 1.48 1.40 1.35 1.31 1.04

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

Tabla 2C. Valores cr´ıticos F(α;ν1 ,ν2 ) de la distribuci´ on F .

21

v.1.00

p = 0.99

α = 0.01 ν1

ν2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 80 100 120 ∞

1 4052. 98.50 34.12 21.20 16.26 13.75 12.25 11.26 10.56 10.04 9.65 9.33 9.07 8.86 8.68 8.53 8.40 8.29 8.18 8.10 8.02 7.95 7.88 7.82 7.77 7.72 7.68 7.64 7.60 7.56 7.31 7.08 6.96 6.90 6.85 6.64

2 4999. 99.00 30.82 18.00 13.27 10.92 9.55 8.65 8.02 7.56 7.21 6.93 6.70 6.51 6.36 6.23 6.11 6.01 5.93 5.85 5.78 5.72 5.66 5.61 5.57 5.53 5.49 5.45 5.42 5.39 5.18 4.98 4.88 4.82 4.79 4.61

3 5403. 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.74 5.56 5.42 5.29 5.18 5.09 5.01 4.94 4.87 4.82 4.76 4.72 4.68 4.64 4.60 4.57 4.54 4.51 4.31 4.13 4.04 3.98 3.95 3.78

4 5625. 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.21 5.04 4.89 4.77 4.67 4.58 4.50 4.43 4.37 4.31 4.26 4.22 4.18 4.14 4.11 4.07 4.04 4.02 3.83 3.65 3.56 3.51 3.48 3.32

5 5764. 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.86 4.69 4.56 4.44 4.34 4.25 4.17 4.10 4.04 3.99 3.94 3.90 3.85 3.82 3.78 3.75 3.73 3.70 3.51 3.34 3.26 3.21 3.17 3.02

6 5859. 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.62 4.46 4.32 4.20 4.10 4.01 3.94 3.87 3.81 3.76 3.71 3.67 3.63 3.59 3.56 3.53 3.50 3.47 3.29 3.12 3.04 2.99 2.96 2.80

7 5928. 99.36 27.67 14.98 10.46 8.26 6.99 6.18 5.61 5.20 4.89 4.64 4.44 4.28 4.14 4.03 3.93 3.84 3.77 3.70 3.64 3.59 3.54 3.50 3.46 3.42 3.39 3.36 3.33 3.30 3.12 2.95 2.87 2.82 2.79 2.64

8 5981. 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.30 4.14 4.00 3.89 3.79 3.71 3.63 3.56 3.51 3.45 3.41 3.36 3.32 3.29 3.26 3.23 3.20 3.17 2.99 2.82 2.74 2.69 2.66 2.51

9 6022. 99.39 27.35 14.66 10.16 7.98 6.72 5.91 5.35 4.94 4.63 4.39 4.19 4.03 3.89 3.78 3.68 3.60 3.52 3.46 3.40 3.35 3.30 3.26 3.22 3.18 3.15 3.12 3.09 3.07 2.89 2.72 2.64 2.59 2.56 2.41

10 6056. 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 4.10 3.94 3.80 3.69 3.59 3.51 3.43 3.37 3.31 3.26 3.21 3.17 3.13 3.09 3.06 3.03 3.00 2.98 2.80 2.63 2.55 2.50 2.47 2.32

12 6106. 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.71 4.40 4.16 3.96 3.80 3.67 3.55 3.46 3.37 3.30 3.23 3.17 3.12 3.07 3.03 2.99 2.96 2.93 2.90 2.87 2.84 2.66 2.50 2.42 2.37 2.34 2.19

15 6157. 99.43 26.87 14.20 9.72 7.56 6.31 5.52 4.96 4.56 4.25 4.01 3.82 3.66 3.52 3.41 3.31 3.23 3.15 3.09 3.03 2.98 2.93 2.89 2.85 2.81 2.78 2.75 2.73 2.70 2.52 2.35 2.27 2.22 2.19 2.04

20 6209. 99.45 26.69 14.02 9.55 7.40 6.16 5.36 4.81 4.41 4.10 3.86 3.66 3.51 3.37 3.26 3.16 3.08 3.00 2.94 2.88 2.83 2.78 2.74 2.70 2.66 2.63 2.60 2.57 2.55 2.37 2.20 2.12 2.07 2.03 1.88

25 6240. 99.46 26.58 13.91 9.45 7.30 6.06 5.26 4.71 4.31 4.01 3.76 3.57 3.41 3.28 3.16 3.07 2.98 2.91 2.84 2.79 2.73 2.69 2.64 2.60 2.57 2.54 2.51 2.48 2.45 2.27 2.10 2.01 1.97 1.93 1.77

50 6303. 99.48 26.35 13.69 9.24 7.09 5.86 5.07 4.52 4.12 3.81 3.57 3.38 3.22 3.08 2.97 2.87 2.78 2.71 2.64 2.58 2.53 2.48 2.44 2.40 2.36 2.33 2.30 2.27 2.25 2.06 1.88 1.79 1.74 1.70 1.53

75 6324. 99.49 26.28 13.61 9.17 7.02 5.79 5.00 4.45 4.05 3.74 3.50 3.31 3.15 3.01 2.90 2.80 2.71 2.64 2.57 2.51 2.46 2.41 2.37 2.33 2.29 2.26 2.23 2.20 2.17 1.98 1.79 1.70 1.65 1.61 1.42

100 6334. 99.49 26.24 13.58 9.13 6.99 5.75 4.96 4.41 4.01 3.71 3.47 3.27 3.11 2.98 2.86 2.76 2.68 2.60 2.54 2.48 2.42 2.37 2.33 2.29 2.25 2.22 2.19 2.16 2.13 1.94 1.75 1.65 1.60 1.56 1.36

∞ 6366. 99.50 26.13 13.46 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.17 3.01 2.87 2.75 2.65 2.57 2.49 2.42 2.36 2.31 2.26 2.21 2.17 2.13 2.10 2.07 2.04 2.01 1.81 1.60 1.50 1.43 1.38 1.05

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

Tabla 2D. Valores cr´ıticos F(α;ν1 ,ν2 ) de la distribuci´ on F .

22

v.1.00

p = 0.995

α = 0.005 ν1

ν2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 80 100 120 ∞

1 16211. 198.50 55.55 31.33 22.78 18.63 16.24 14.69 13.61 12.83 12.23 11.75 11.37 11.06 10.80 10.58 10.38 10.22 10.07 9.94 9.83 9.73 9.63 9.55 9.48 9.41 9.34 9.28 9.23 9.18 8.83 8.49 8.33 8.24 8.18 7.88

2 19999. 199.00 49.80 26.28 18.31 14.54 12.40 11.04 10.11 9.43 8.91 8.51 8.19 7.92 7.70 7.51 7.35 7.21 7.09 6.99 6.89 6.81 6.73 6.66 6.60 6.54 6.49 6.44 6.40 6.35 6.07 5.79 5.67 5.59 5.54 5.30

3 21615. 199.17 47.47 24.26 16.53 12.92 10.88 9.60 8.72 8.08 7.60 7.23 6.93 6.68 6.48 6.30 6.16 6.03 5.92 5.82 5.73 5.65 5.58 5.52 5.46 5.41 5.36 5.32 5.28 5.24 4.98 4.73 4.61 4.54 4.50 4.28

4 22500. 199.25 46.19 23.15 15.56 12.03 10.05 8.81 7.96 7.34 6.88 6.52 6.23 6.00 5.80 5.64 5.50 5.37 5.27 5.17 5.09 5.02 4.95 4.89 4.84 4.79 4.74 4.70 4.66 4.62 4.37 4.14 4.03 3.96 3.92 3.72

5 23056. 199.30 45.39 22.46 14.94 11.46 9.52 8.30 7.47 6.87 6.42 6.07 5.79 5.56 5.37 5.21 5.07 4.96 4.85 4.76 4.68 4.61 4.54 4.49 4.43 4.38 4.34 4.30 4.26 4.23 3.99 3.76 3.65 3.59 3.55 3.35

6 23437. 199.33 44.84 21.97 14.51 11.07 9.16 7.95 7.13 6.54 6.10 5.76 5.48 5.26 5.07 4.91 4.78 4.66 4.56 4.47 4.39 4.32 4.26 4.20 4.15 4.10 4.06 4.02 3.98 3.95 3.71 3.49 3.39 3.33 3.28 3.09

7 23715. 199.36 44.43 21.62 14.20 10.79 8.89 7.69 6.88 6.30 5.86 5.52 5.25 5.03 4.85 4.69 4.56 4.44 4.34 4.26 4.18 4.11 4.05 3.99 3.94 3.89 3.85 3.81 3.77 3.74 3.51 3.29 3.19 3.13 3.09 2.90

8 23925. 199.37 44.13 21.35 13.96 10.57 8.68 7.50 6.69 6.12 5.68 5.35 5.08 4.86 4.67 4.52 4.39 4.28 4.18 4.09 4.01 3.94 3.88 3.83 3.78 3.73 3.69 3.65 3.61 3.58 3.35 3.13 3.03 2.97 2.93 2.75

9 24091. 199.39 43.88 21.14 13.77 10.39 8.51 7.34 6.54 5.97 5.54 5.20 4.94 4.72 4.54 4.38 4.25 4.14 4.04 3.96 3.88 3.81 3.75 3.69 3.64 3.60 3.56 3.52 3.48 3.45 3.22 3.01 2.91 2.85 2.81 2.62

10 24224. 199.40 43.69 20.97 13.62 10.25 8.38 7.21 6.42 5.85 5.42 5.09 4.82 4.60 4.42 4.27 4.14 4.03 3.93 3.85 3.77 3.70 3.64 3.59 3.54 3.49 3.45 3.41 3.38 3.34 3.12 2.90 2.80 2.74 2.71 2.52

12 24426. 199.42 43.39 20.70 13.38 10.03 8.18 7.01 6.23 5.66 5.24 4.91 4.64 4.43 4.25 4.10 3.97 3.86 3.76 3.68 3.60 3.54 3.47 3.42 3.37 3.33 3.28 3.25 3.21 3.18 2.95 2.74 2.64 2.58 2.54 2.36

15 24630. 199.43 43.08 20.44 13.15 9.81 7.97 6.81 6.03 5.47 5.05 4.72 4.46 4.25 4.07 3.92 3.79 3.68 3.59 3.50 3.43 3.36 3.30 3.25 3.20 3.15 3.11 3.07 3.04 3.01 2.78 2.57 2.47 2.41 2.37 2.19

20 24836. 199.45 42.78 20.17 12.90 9.59 7.75 6.61 5.83 5.27 4.86 4.53 4.27 4.06 3.88 3.73 3.61 3.50 3.40 3.32 3.24 3.18 3.12 3.06 3.01 2.97 2.93 2.89 2.86 2.82 2.60 2.39 2.29 2.23 2.19 2.00

25 24960. 199.46 42.59 20.00 12.76 9.45 7.62 6.48 5.71 5.15 4.74 4.41 4.15 3.94 3.77 3.62 3.49 3.38 3.29 3.20 3.13 3.06 3.00 2.95 2.90 2.85 2.81 2.77 2.74 2.71 2.48 2.27 2.17 2.11 2.07 1.88

50 25211. 199.48 42.21 19.67 12.45 9.17 7.35 6.22 5.45 4.90 4.49 4.17 3.91 3.70 3.52 3.37 3.25 3.14 3.04 2.96 2.88 2.82 2.76 2.70 2.65 2.61 2.57 2.53 2.49 2.46 2.23 2.01 1.90 1.84 1.80 1.59

75 25295. 199.49 42.09 19.55 12.35 9.07 7.26 6.13 5.37 4.82 4.40 4.08 3.82 3.61 3.44 3.29 3.16 3.05 2.96 2.87 2.80 2.73 2.67 2.61 2.56 2.52 2.48 2.44 2.40 2.37 2.14 1.91 1.80 1.74 1.69 1.47

100 25337. 199.49 42.02 19.50 12.30 9.03 7.22 6.09 5.32 4.77 4.36 4.04 3.78 3.57 3.39 3.25 3.12 3.01 2.91 2.83 2.75 2.69 2.62 2.57 2.52 2.47 2.43 2.39 2.36 2.32 2.09 1.86 1.75 1.68 1.64 1.40

∞ 25463. 199.50 41.83 19.33 12.15 8.88 7.08 5.95 5.19 4.64 4.23 3.91 3.65 3.44 3.26 3.11 2.99 2.87 2.78 2.69 2.62 2.55 2.49 2.43 2.38 2.33 2.29 2.25 2.21 2.18 1.93 1.69 1.57 1.49 1.43 1.05

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

Tabla 2E. Valores cr´ıticos F(α;ν1 ,ν2 ) de la distribuci´ on F .

23

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

2.8.

24

Distribuci´ on del estad´ıstico d de Durbin-Watson

Se define el estadstico de Durbin-Watson Pn (zi − zi−1 )2 , d = i=2Pn 2 i=1 zi

donde zi son los residuales del modelo lineal

H0 : ρ = 0 vs. H0 : ρ ≠ 0 Inconclusa Rechazar

Aceptar dU

dL

zi = yi − βˆ0 − βˆ1 xi1 − . . . − βˆk xik , con 1 ≤ i ≤ n.

d

Prueba para autocorrelación positiva ρ > 0 de significancia α: Si Si Si

d < dL(α;n,k) d > dU(α;n,k) dL(α;n,k) < d < dU(α;n,k)

Hay evidencia significativa de autocorrelación positiva No hay evidencia significativa de autocorrelación positiva La prueba es inconcluyente

Prueba para autocorrelación negativa ρ < 0 de significancia α: Si Si Si

4 − d < dL(α;n,k) 4 − d > dU(α;n,k) dL(α;n,k) < 4 − d < dU(α;n,k)

Hay evidencia significativa de autocorrelación negativa No hay evidencia significativa de autocorrelación negativa La prueba es inconcluyente

Prueba de dos colas para autocorrelación |ρ| > 0 de significancia α: Si Si

´ 4 − d < dL( α2 ;n,k) d < dL( α2 ;n,k) o o 4 − d > dU( α2 ;n,k) d > dU( α2 ;n,k) ´ En otro caso

c E. Barrios y J. A. Garc´ıa

Hay evidencia significativa de autocorrelación No hay evidencia significativa de autocorrelación La prueba es inconcluyente

v.1.00

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.01 k = 1 dL dU 0.390 1.142 0.435 1.036 0.498 1.004 0.554 0.998 0.604 1.000 0.653 1.010 0.698 1.023 0.739 1.039 0.776 1.054 0.811 1.071 0.844 1.087 0.874 1.103 0.902 1.118 0.928 1.132 0.953 1.147 0.976 1.161 0.997 1.174 1.017 1.186 1.037 1.199 1.055 1.211 1.072 1.222 1.088 1.233 1.104 1.244 1.119 1.254 1.133 1.264 1.147 1.273 1.159 1.281 1.172 1.290 1.184 1.299 1.195 1.307 1.207 1.315 1.217 1.322 1.227 1.331 1.237 1.337 1.246 1.344 1.255 1.351 1.264 1.357 1.272 1.363 1.280 1.369 1.288 1.376 1.296 1.382 1.303 1.387 1.310 1.392 1.318 1.398 1.324 1.403 1.355 1.427 1.383 1.449 1.408 1.468 1.428 1.485 1.449 1.501 1.466 1.515 1.482 1.528 1.497 1.541 1.510 1.552 1.523 1.562 1.573 1.605 1.611 1.637 1.640 1.663 1.664 1.684

k = 2 dL dU · · 0.294 1.676 0.346 1.489 0.409 1.390 0.466 1.332 0.519 1.297 0.570 1.274 0.616 1.261 0.660 1.254 0.700 1.251 0.738 1.253 0.773 1.256 0.805 1.260 0.835 1.265 0.863 1.270 0.890 1.278 0.915 1.284 0.938 1.290 0.960 1.298 0.981 1.305 1.001 1.312 1.019 1.318 1.037 1.325 1.054 1.332 1.070 1.339 1.086 1.346 1.100 1.352 1.114 1.358 1.127 1.364 1.140 1.370 1.153 1.376 1.165 1.382 1.176 1.388 1.187 1.393 1.198 1.399 1.208 1.404 1.218 1.409 1.227 1.414 1.236 1.418 1.245 1.423 1.253 1.428 1.262 1.433 1.270 1.437 1.278 1.441 1.285 1.446 1.320 1.466 1.350 1.484 1.377 1.500 1.401 1.515 1.422 1.529 1.441 1.541 1.458 1.553 1.474 1.563 1.489 1.573 1.502 1.582 1.558 1.622 1.598 1.651 1.629 1.674 1.654 1.694

k = 3 dL dU · · · · 0.229 2.102 0.279 1.875 0.340 1.734 0.396 1.641 0.448 1.574 0.499 1.526 0.547 1.491 0.592 1.465 0.632 1.445 0.672 1.433 0.708 1.423 0.742 1.416 0.773 1.410 0.803 1.408 0.832 1.408 0.857 1.406 0.882 1.407 0.906 1.409 0.928 1.410 0.950 1.413 0.969 1.415 0.988 1.418 1.006 1.422 1.024 1.425 1.040 1.429 1.055 1.432 1.070 1.435 1.085 1.439 1.098 1.442 1.112 1.446 1.125 1.450 1.137 1.453 1.148 1.456 1.160 1.460 1.170 1.464 1.181 1.467 1.191 1.471 1.201 1.474 1.210 1.477 1.219 1.481 1.228 1.484 1.237 1.488 1.246 1.491 1.283 1.506 1.316 1.520 1.346 1.534 1.372 1.546 1.394 1.557 1.416 1.568 1.435 1.578 1.452 1.587 1.468 1.596 1.482 1.604 1.541 1.638 1.585 1.665 1.617 1.686 1.644 1.704

k = 4 dL dU · · · · · · 0.183 2.433 0.230 2.194 0.286 2.029 0.339 1.912 0.391 1.826 0.441 1.757 0.487 1.704 0.532 1.663 0.574 1.631 0.613 1.604 0.650 1.584 0.685 1.567 0.718 1.554 0.748 1.543 0.778 1.535 0.804 1.528 0.831 1.523 0.855 1.519 0.879 1.516 0.900 1.513 0.922 1.512 0.942 1.510 0.960 1.510 0.979 1.510 0.996 1.510 1.012 1.511 1.028 1.512 1.044 1.513 1.059 1.515 1.072 1.516 1.086 1.517 1.098 1.518 1.111 1.520 1.123 1.522 1.135 1.524 1.146 1.526 1.156 1.528 1.167 1.530 1.177 1.531 1.187 1.534 1.196 1.536 1.205 1.538 1.247 1.549 1.283 1.558 1.315 1.569 1.343 1.578 1.367 1.586 1.390 1.595 1.411 1.603 1.429 1.611 1.446 1.618 1.462 1.625 1.525 1.655 1.571 1.679 1.606 1.698 1.634 1.714

k = 5 dL dU · · · · · · · · 0.150 2.690 0.193 2.453 0.244 2.280 0.294 2.149 0.343 2.049 0.391 1.967 0.437 1.900 0.480 1.846 0.522 1.803 0.561 1.767 0.598 1.737 0.634 1.712 0.667 1.691 0.698 1.673 0.728 1.658 0.756 1.646 0.783 1.635 0.808 1.626 0.832 1.618 0.855 1.611 0.877 1.606 0.897 1.601 0.917 1.598 0.936 1.594 0.954 1.591 0.972 1.589 0.988 1.587 1.004 1.586 1.019 1.585 1.034 1.585 1.048 1.584 1.062 1.584 1.074 1.583 1.087 1.583 1.100 1.584 1.111 1.584 1.122 1.584 1.133 1.585 1.144 1.586 1.155 1.587 1.164 1.587 1.209 1.592 1.249 1.598 1.283 1.605 1.314 1.611 1.340 1.617 1.364 1.624 1.386 1.630 1.406 1.636 1.425 1.642 1.441 1.647 1.509 1.673 1.557 1.693 1.594 1.710 1.623 1.725

k = 6 dL dU · · · · · · · · · · 0.125 2.893 0.164 2.664 0.211 2.490 0.257 2.354 0.303 2.244 0.349 2.153 0.393 2.078 0.435 2.016 0.476 1.962 0.515 1.919 0.552 1.881 0.587 1.848 0.620 1.821 0.652 1.797 0.682 1.777 0.711 1.759 0.738 1.744 0.764 1.729 0.788 1.717 0.812 1.707 0.834 1.698 0.856 1.690 0.876 1.683 0.895 1.677 0.914 1.671 0.932 1.666 0.950 1.662 0.966 1.658 0.982 1.655 0.997 1.652 1.012 1.650 1.026 1.648 1.040 1.646 1.053 1.644 1.066 1.643 1.078 1.642 1.090 1.641 1.101 1.640 1.112 1.639 1.123 1.639 1.172 1.638 1.214 1.640 1.251 1.642 1.284 1.645 1.313 1.649 1.339 1.653 1.362 1.657 1.384 1.662 1.403 1.666 1.420 1.670 1.492 1.690 1.543 1.707 1.582 1.722 1.613 1.735

k = 7 dL dU · · · · · · · · · · · · 0.105 3.053 0.140 2.838 0.183 2.667 0.226 2.530 0.269 2.415 0.313 2.319 0.355 2.237 0.396 2.169 0.436 2.109 0.474 2.060 0.510 2.015 0.545 1.978 0.578 1.944 0.610 1.915 0.641 1.890 0.669 1.867 0.696 1.847 0.723 1.829 0.748 1.814 0.772 1.799 0.794 1.787 0.817 1.776 0.837 1.766 0.858 1.757 0.876 1.749 0.895 1.742 0.913 1.736 0.930 1.729 0.946 1.724 0.962 1.719 0.977 1.715 0.992 1.711 1.006 1.707 1.019 1.704 1.033 1.702 1.046 1.699 1.058 1.697 1.070 1.694 1.081 1.692 1.134 1.686 1.179 1.682 1.218 1.680 1.253 1.680 1.284 1.681 1.312 1.683 1.338 1.685 1.359 1.687 1.380 1.690 1.399 1.693 1.475 1.708 1.529 1.722 1.570 1.735 1.602 1.746

k = 8 dL dU · · · · · · · · · · · · · · 0.090 3.182 0.122 2.982 0.161 2.817 0.200 2.681 0.241 2.566 0.282 2.466 0.322 2.382 0.362 2.307 0.400 2.244 0.437 2.189 0.473 2.140 0.507 2.098 0.540 2.060 0.571 2.026 0.602 1.997 0.631 1.971 0.658 1.947 0.684 1.925 0.710 1.906 0.734 1.890 0.757 1.874 0.779 1.860 0.800 1.847 0.821 1.836 0.840 1.825 0.859 1.815 0.878 1.807 0.895 1.799 0.912 1.792 0.928 1.785 0.944 1.779 0.959 1.773 0.974 1.768 0.988 1.764 1.001 1.759 1.014 1.755 1.027 1.751 1.039 1.747 1.095 1.734 1.143 1.725 1.186 1.720 1.223 1.716 1.256 1.714 1.285 1.714 1.312 1.714 1.336 1.714 1.359 1.716 1.378 1.717 1.458 1.726 1.515 1.737 1.558 1.747 1.592 1.757

k = 9 dL dU · · · · · · · · · · · · · · · · 0.078 3.287 0.107 3.101 0.142 2.944 0.179 2.811 0.217 2.697 0.255 2.597 0.293 2.510 0.332 2.434 0.369 2.367 0.404 2.308 0.439 2.255 0.473 2.209 0.505 2.168 0.536 2.131 0.566 2.097 0.595 2.068 0.622 2.041 0.649 2.017 0.674 1.995 0.698 1.975 0.722 1.957 0.744 1.940 0.766 1.925 0.787 1.912 0.807 1.899 0.826 1.887 0.845 1.877 0.862 1.866 0.880 1.857 0.896 1.849 0.912 1.841 0.927 1.834 0.942 1.827 0.957 1.822 0.970 1.815 0.984 1.810 0.997 1.805 1.057 1.785 1.108 1.771 1.153 1.761 1.192 1.753 1.228 1.749 1.259 1.745 1.287 1.743 1.312 1.741 1.336 1.741 1.357 1.741 1.441 1.745 1.501 1.752 1.546 1.760 1.582 1.768

k = dL · · · · · · · · · 0.068 0.094 0.127 0.160 0.195 0.232 0.268 0.305 0.340 0.375 0.409 0.441 0.473 0.504 0.534 0.562 0.590 0.615 0.641 0.665 0.688 0.711 0.733 0.754 0.774 0.794 0.813 0.830 0.848 0.865 0.881 0.897 0.912 0.927 0.941 0.955 1.018 1.072 1.120 1.162 1.199 1.232 1.261 1.289 1.313 1.335 1.424 1.487 1.534 1.571

10 dU · · · · · · · · · 3.374 3.201 3.053 2.924 2.813 2.714 2.625 2.547 2.479 2.417 2.362 2.313 2.269 2.229 2.193 2.160 2.131 2.104 2.079 2.057 2.037 2.017 2.000 1.985 1.970 1.957 1.944 1.932 1.921 1.911 1.902 1.893 1.885 1.878 1.871 1.864 1.837 1.817 1.803 1.792 1.784 1.777 1.772 1.769 1.767 1.765 1.763 1.767 1.773 1.779

k = dL · · · · · · · · · · 0.060 0.084 0.114 0.144 0.178 0.212 0.246 0.281 0.315 0.348 0.381 0.413 0.444 0.474 0.503 0.531 0.558 0.585 0.610 0.634 0.657 0.681 0.702 0.723 0.744 0.764 0.782 0.800 0.818 0.835 0.852 0.868 0.884 0.898 0.913 0.979 1.036 1.086 1.130 1.170 1.205 1.235 1.264 1.290 1.314 1.407 1.472 1.522 1.561

11 dU · · · · · · · · · · 3.446 3.286 3.146 3.023 2.915 2.816 2.729 2.650 2.580 2.518 2.461 2.410 2.363 2.321 2.282 2.248 2.216 2.187 2.160 2.136 2.113 2.093 2.072 2.055 2.039 2.024 2.010 1.996 1.984 1.972 1.962 1.951 1.942 1.933 1.926 1.890 1.865 1.845 1.831 1.819 1.810 1.803 1.797 1.794 1.790 1.783 1.782 1.786 1.791

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2A.1 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

25

v.1.00

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.01 k = 12 dL dU 0.053 3.506 0.075 3.358 0.102 3.227 0.131 3.109 0.162 3.004 0.194 2.909 0.227 2.822 0.260 2.744 0.292 2.674 0.325 2.609 0.356 2.551 0.387 2.499 0.418 2.451 0.447 2.407 0.475 2.367 0.503 2.330 0.530 2.296 0.556 2.265 0.581 2.237 0.605 2.210 0.628 2.187 0.651 2.164 0.673 2.143 0.694 2.123 0.715 2.105 0.734 2.088 0.753 2.073 0.771 2.058 0.790 2.044 0.807 2.031 0.824 2.020 0.840 2.008 0.856 1.998 0.871 1.987 0.940 1.945 1.001 1.914 1.054 1.890 1.099 1.870 1.141 1.856 1.177 1.844 1.211 1.835 1.240 1.826 1.267 1.821 1.292 1.815 1.389 1.802 1.458 1.798 1.510 1.799 1.550 1.801

k = dL · 0.047 0.067 0.093 0.119 0.148 0.178 0.209 0.241 0.272 0.303 0.333 0.364 0.393 0.422 0.450 0.477 0.503 0.529 0.554 0.577 0.601 0.623 0.646 0.666 0.687 0.707 0.726 0.744 0.762 0.780 0.797 0.813 0.829 0.902 0.964 1.020 1.068 1.111 1.150 1.184 1.216 1.243 1.270 1.372 1.444 1.497 1.539

13 dU · 3.557 3.420 3.297 3.185 3.084 2.991 2.906 2.829 2.759 2.694 2.635 2.582 2.532 2.487 2.446 2.407 2.373 2.340 2.310 2.282 2.256 2.231 2.210 2.189 2.169 2.150 2.134 2.118 2.103 2.089 2.076 2.063 2.051 2.002 1.963 1.935 1.911 1.893 1.879 1.866 1.857 1.848 1.841 1.822 1.814 1.812 1.813

k = dL · · 0.043 0.061 0.084 0.109 0.136 0.165 0.194 0.224 0.254 0.283 0.313 0.342 0.370 0.398 0.426 0.452 0.478 0.504 0.528 0.552 0.575 0.598 0.619 0.641 0.661 0.680 0.699 0.718 0.736 0.754 0.771 0.787 0.863 0.929 0.986 1.037 1.082 1.122 1.158 1.191 1.220 1.248 1.355 1.429 1.485 1.529

14 dU · · 3.601 3.474 3.358 3.252 3.155 3.065 2.982 2.906 2.836 2.772 2.713 2.659 2.609 2.562 2.520 2.481 2.444 2.410 2.379 2.350 2.323 2.297 2.273 2.251 2.230 2.211 2.192 2.176 2.159 2.144 2.130 2.116 2.058 2.015 1.981 1.953 1.931 1.913 1.899 1.886 1.876 1.868 1.842 1.830 1.826 1.824

k = dL · · · 0.038 0.055 0.077 0.100 0.125 0.152 0.180 0.208 0.237 0.265 0.294 0.322 0.350 0.377 0.404 0.430 0.456 0.480 0.504 0.528 0.551 0.573 0.594 0.616 0.636 0.655 0.675 0.694 0.712 0.729 0.746 0.824 0.893 0.952 1.005 1.052 1.094 1.132 1.166 1.197 1.225 1.337 1.414 1.473 1.518

15 dU · · · 3.639 3.521 3.412 3.311 3.218 3.131 3.051 2.976 2.907 2.843 2.785 2.730 2.680 2.633 2.589 2.549 2.512 2.478 2.445 2.414 2.386 2.360 2.335 2.312 2.289 2.269 2.250 2.231 2.214 2.198 2.183 2.117 2.067 2.027 1.995 1.969 1.949 1.932 1.917 1.905 1.895 1.862 1.846 1.840 1.836

k = dL · · · · 0.035 0.050 0.070 0.092 0.116 0.141 0.167 0.194 0.222 0.249 0.277 0.304 0.331 0.357 0.383 0.409 0.433 0.458 0.482 0.505 0.528 0.550 0.571 0.592 0.612 0.632 0.651 0.670 0.688 0.705 0.786 0.857 0.919 0.974 1.023 1.067 1.106 1.141 1.173 1.203 1.319 1.400 1.460 1.506

16 dU · · · · 3.671 3.562 3.459 3.363 3.274 3.191 3.112 3.040 2.972 2.909 2.851 2.796 2.746 2.699 2.655 2.614 2.576 2.541 2.507 2.476 2.447 2.419 2.393 2.369 2.346 2.325 2.305 2.286 2.267 2.250 2.176 2.119 2.075 2.038 2.009 1.985 1.965 1.948 1.934 1.922 1.882 1.863 1.853 1.847

k = dL · · · · · 0.032 0.046 0.065 0.085 0.107 0.131 0.156 0.182 0.208 0.234 0.261 0.287 0.313 0.339 0.364 0.390 0.414 0.438 0.461 0.484 0.506 0.528 0.549 0.569 0.590 0.609 0.628 0.647 0.665 0.748 0.822 0.886 0.943 0.993 1.039 1.080 1.116 1.150 1.181 1.301 1.385 1.447 1.496

17 dU · · · · · 3.700 3.598 3.501 3.410 3.326 3.245 3.169 3.098 3.032 2.970 2.912 2.858 2.808 2.761 2.717 2.675 2.637 2.600 2.566 2.534 2.504 2.476 2.450 2.424 2.401 2.378 2.357 2.338 2.318 2.236 2.173 2.123 2.082 2.049 2.022 1.999 1.979 1.963 1.949 1.903 1.880 1.867 1.860

k = dL · · · · · · 0.029 0.042 0.060 0.079 0.100 0.122 0.146 0.171 0.196 0.221 0.246 0.272 0.297 0.322 0.346 0.371 0.395 0.418 0.441 0.464 0.486 0.507 0.528 0.548 0.569 0.588 0.607 0.625 0.711 0.787 0.852 0.911 0.964 1.011 1.053 1.091 1.126 1.158 1.283 1.370 1.435 1.485

18 dU · · · · · · 3.725 3.629 3.538 3.452 3.371 3.294 3.220 3.152 3.086 3.026 2.969 2.915 2.865 2.819 2.774 2.733 2.694 2.657 2.623 2.590 2.559 2.530 2.503 2.477 2.453 2.430 2.408 2.387 2.298 2.228 2.171 2.126 2.090 2.059 2.033 2.011 1.993 1.978 1.925 1.897 1.881 1.872

k = dL · · · · · · · 0.027 0.039 0.055 0.073 0.093 0.114 0.137 0.160 0.184 0.209 0.233 0.257 0.282 0.306 0.330 0.354 0.377 0.400 0.422 0.445 0.467 0.487 0.508 0.528 0.548 0.568 0.586 0.674 0.751 0.819 0.880 0.934 0.983 1.027 1.066 1.103 1.135 1.265 1.355 1.422 1.473

19 dU · · · · · · · 3.747 3.657 3.572 3.490 3.412 3.338 3.268 3.201 3.137 3.078 3.022 2.969 2.919 2.872 2.828 2.786 2.747 2.711 2.676 2.643 2.612 2.582 2.554 2.528 2.503 2.479 2.456 2.359 2.283 2.222 2.172 2.131 2.096 2.069 2.044 2.024 2.005 1.946 1.913 1.895 1.883

k = dL · · · · · · · · 0.025 0.036 0.051 0.068 0.087 0.107 0.129 0.151 0.174 0.197 0.221 0.244 0.268 0.292 0.315 0.338 0.361 0.383 0.405 0.427 0.448 0.469 0.490 0.509 0.529 0.548 0.637 0.716 0.786 0.849 0.905 0.955 1.001 1.041 1.079 1.113 1.246 1.340 1.409 1.462

20 dU · · · · · · · · 3.766 3.682 3.602 3.524 3.450 3.379 3.310 3.246 3.184 3.126 3.071 3.019 2.969 2.923 2.879 2.837 2.799 2.761 2.726 2.693 2.662 2.631 2.603 2.576 2.551 2.526 2.421 2.338 2.272 2.217 2.172 2.135 2.104 2.076 2.054 2.034 1.967 1.931 1.909 1.896

k = dL · · · · · · · · · · · · · 0.017 0.026 0.037 0.049 0.064 0.079 0.096 0.114 0.133 0.151 0.171 0.191 0.211 0.232 0.252 0.272 0.292 0.312 0.332 0.351 0.371 0.464 0.549 0.626 0.696 0.759 0.817 0.869 0.916 0.959 0.999 1.155 1.263 1.344 1.406

25 dU · · · · · · · · · · · · · 3.836 3.775 3.713 3.653 3.594 3.535 3.479 3.424 3.370 3.319 3.270 3.223 3.177 3.134 3.092 3.052 3.014 2.978 2.943 2.909 2.878 2.736 2.623 2.530 2.452 2.387 2.333 2.287 2.246 2.212 2.183 2.079 2.019 1.982 1.958

k = dL · · · · · · · · · · · · · · · · · · 0.013 0.019 0.028 0.037 0.049 0.061 0.075 0.089 0.104 0.120 0.137 0.153 0.170 0.188 0.205 0.223 0.311 0.397 0.477 0.551 0.620 0.682 0.740 0.793 0.841 0.885 1.062 1.186 1.278 1.349

30 dU · · · · · · · · · · · · · · · · · · 3.879 3.832 3.784 3.736 3.688 3.640 3.593 3.547 3.501 3.456 3.413 3.371 3.330 3.291 3.252 3.215 3.048 2.908 2.791 2.693 2.609 2.538 2.477 2.424 2.378 2.338 2.196 2.112 2.059 2.023

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2A.2 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

26

v.1.00

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.025 k = 1 dL dU 0.489 1.258 0.564 1.191 0.633 1.172 0.690 1.164 0.744 1.165 0.794 1.173 0.838 1.183 0.879 1.196 0.916 1.209 0.949 1.222 0.980 1.236 1.009 1.249 1.035 1.262 1.060 1.274 1.083 1.286 1.104 1.297 1.124 1.309 1.143 1.319 1.161 1.329 1.177 1.339 1.193 1.348 1.208 1.358 1.223 1.367 1.236 1.375 1.249 1.383 1.261 1.391 1.272 1.398 1.284 1.406 1.295 1.413 1.305 1.420 1.315 1.426 1.324 1.433 1.333 1.439 1.342 1.445 1.350 1.451 1.359 1.457 1.366 1.462 1.373 1.467 1.381 1.472 1.388 1.477 1.395 1.482 1.401 1.487 1.408 1.491 1.414 1.496 1.420 1.500 1.447 1.520 1.471 1.538 1.493 1.554 1.511 1.568 1.528 1.582 1.544 1.594 1.557 1.605 1.570 1.615 1.582 1.624 1.593 1.633 1.637 1.669 1.669 1.696 1.694 1.717 1.714 1.734

k = 2 dL dU · · 0.372 1.775 0.450 1.629 0.520 1.547 0.581 1.493 0.640 1.456 0.694 1.433 0.742 1.418 0.787 1.409 0.827 1.405 0.864 1.403 0.899 1.403 0.930 1.405 0.959 1.407 0.987 1.411 1.013 1.415 1.036 1.420 1.059 1.424 1.080 1.429 1.099 1.434 1.118 1.439 1.136 1.445 1.152 1.450 1.168 1.455 1.183 1.460 1.197 1.465 1.211 1.469 1.223 1.474 1.236 1.479 1.247 1.483 1.259 1.488 1.270 1.493 1.280 1.497 1.291 1.502 1.300 1.506 1.309 1.510 1.319 1.514 1.327 1.518 1.335 1.522 1.343 1.525 1.351 1.529 1.358 1.533 1.366 1.536 1.373 1.540 1.380 1.543 1.411 1.559 1.438 1.574 1.462 1.587 1.482 1.598 1.501 1.610 1.519 1.620 1.534 1.629 1.548 1.638 1.561 1.646 1.572 1.653 1.621 1.685 1.656 1.710 1.683 1.729 1.705 1.745

k = 3 dL dU · · · · 0.291 2.184 0.365 1.997 0.434 1.878 0.496 1.791 0.555 1.725 0.609 1.677 0.660 1.642 0.707 1.615 0.748 1.594 0.788 1.578 0.825 1.567 0.859 1.558 0.890 1.551 0.920 1.546 0.948 1.544 0.973 1.540 0.997 1.539 1.020 1.539 1.041 1.538 1.061 1.539 1.080 1.540 1.098 1.541 1.116 1.543 1.132 1.544 1.148 1.547 1.162 1.548 1.176 1.550 1.190 1.553 1.202 1.555 1.215 1.557 1.227 1.560 1.238 1.562 1.249 1.564 1.260 1.567 1.270 1.569 1.279 1.572 1.289 1.574 1.298 1.576 1.306 1.579 1.315 1.581 1.323 1.584 1.331 1.586 1.339 1.589 1.373 1.600 1.403 1.610 1.430 1.620 1.453 1.630 1.474 1.638 1.493 1.647 1.510 1.654 1.525 1.662 1.540 1.669 1.552 1.675 1.604 1.702 1.642 1.723 1.671 1.741 1.694 1.755

k = 4 dL dU · · · · · · 0.234 2.502 0.302 2.300 0.366 2.158 0.425 2.050 0.484 1.966 0.538 1.900 0.589 1.848 0.636 1.806 0.680 1.773 0.720 1.746 0.758 1.723 0.794 1.705 0.827 1.691 0.857 1.677 0.887 1.668 0.914 1.659 0.939 1.651 0.963 1.646 0.986 1.641 1.008 1.638 1.028 1.635 1.047 1.631 1.065 1.630 1.083 1.628 1.100 1.627 1.116 1.626 1.131 1.626 1.145 1.626 1.159 1.626 1.172 1.626 1.185 1.626 1.197 1.626 1.209 1.627 1.220 1.628 1.231 1.628 1.242 1.629 1.252 1.631 1.262 1.632 1.271 1.633 1.280 1.634 1.288 1.635 1.297 1.636 1.336 1.642 1.369 1.649 1.398 1.655 1.423 1.662 1.446 1.668 1.467 1.674 1.485 1.680 1.502 1.686 1.517 1.691 1.532 1.696 1.588 1.719 1.629 1.738 1.660 1.753 1.684 1.766

k = 5 dL dU · · · · · · · · 0.191 2.749 0.253 2.545 0.312 2.394 0.369 2.275 0.425 2.178 0.478 2.099 0.528 2.036 0.574 1.982 0.619 1.939 0.660 1.902 0.699 1.872 0.735 1.845 0.769 1.823 0.801 1.804 0.831 1.788 0.859 1.774 0.886 1.761 0.911 1.751 0.935 1.742 0.957 1.733 0.979 1.727 0.998 1.720 1.018 1.715 1.037 1.711 1.054 1.707 1.071 1.703 1.087 1.700 1.102 1.697 1.117 1.695 1.131 1.693 1.145 1.692 1.158 1.690 1.170 1.689 1.182 1.688 1.194 1.688 1.205 1.687 1.215 1.686 1.226 1.686 1.236 1.686 1.245 1.686 1.255 1.685 1.297 1.686 1.334 1.689 1.365 1.691 1.394 1.695 1.418 1.699 1.440 1.703 1.460 1.707 1.479 1.711 1.495 1.715 1.511 1.718 1.571 1.736 1.615 1.752 1.648 1.765 1.674 1.776

k = 6 dL dU · · · · · · · · · · 0.160 2.943 0.215 2.745 0.269 2.592 0.322 2.468 0.376 2.363 0.427 2.276 0.475 2.204 0.521 2.143 0.565 2.091 0.606 2.048 0.645 2.010 0.681 1.977 0.716 1.948 0.748 1.923 0.779 1.902 0.808 1.883 0.836 1.867 0.862 1.851 0.886 1.838 0.910 1.827 0.932 1.816 0.953 1.807 0.973 1.799 0.992 1.791 1.011 1.785 1.028 1.778 1.045 1.773 1.061 1.768 1.077 1.764 1.091 1.760 1.106 1.756 1.119 1.753 1.132 1.750 1.145 1.748 1.157 1.746 1.169 1.743 1.180 1.741 1.191 1.740 1.201 1.738 1.212 1.737 1.258 1.732 1.298 1.730 1.332 1.729 1.363 1.729 1.390 1.730 1.414 1.732 1.436 1.734 1.455 1.736 1.473 1.739 1.490 1.741 1.554 1.754 1.601 1.766 1.636 1.777 1.664 1.787

k = 7 dL dU · · · · · · · · · · · · 0.135 3.096 0.185 2.909 0.234 2.758 0.284 2.633 0.334 2.525 0.383 2.433 0.430 2.355 0.475 2.289 0.517 2.231 0.558 2.182 0.597 2.138 0.633 2.100 0.668 2.067 0.701 2.037 0.732 2.011 0.761 1.987 0.789 1.967 0.816 1.948 0.841 1.932 0.865 1.916 0.888 1.903 0.910 1.891 0.931 1.880 0.950 1.870 0.970 1.860 0.987 1.852 1.005 1.845 1.022 1.837 1.038 1.831 1.053 1.825 1.068 1.819 1.082 1.815 1.096 1.810 1.109 1.806 1.122 1.803 1.135 1.799 1.146 1.796 1.157 1.793 1.168 1.790 1.219 1.780 1.262 1.772 1.299 1.767 1.332 1.764 1.361 1.763 1.387 1.762 1.411 1.762 1.431 1.762 1.451 1.763 1.468 1.764 1.538 1.772 1.587 1.781 1.624 1.789 1.653 1.797

k = 8 dL dU · · · · · · · · · · · · · · 0.116 3.220 0.160 3.044 0.206 2.899 0.252 2.775 0.299 2.666 0.345 2.572 0.390 2.491 0.433 2.420 0.475 2.359 0.515 2.305 0.553 2.257 0.589 2.215 0.624 2.177 0.657 2.144 0.688 2.114 0.718 2.087 0.746 2.063 0.773 2.040 0.798 2.020 0.823 2.003 0.847 1.987 0.869 1.972 0.890 1.957 0.910 1.945 0.930 1.934 0.949 1.923 0.967 1.914 0.984 1.905 1.001 1.897 1.017 1.889 1.032 1.882 1.047 1.876 1.061 1.870 1.075 1.864 1.088 1.859 1.101 1.854 1.113 1.850 1.125 1.845 1.179 1.828 1.225 1.815 1.266 1.807 1.301 1.801 1.332 1.796 1.360 1.793 1.385 1.790 1.407 1.789 1.428 1.788 1.447 1.788 1.520 1.790 1.572 1.796 1.612 1.802 1.642 1.808

k = 9 dL dU · · · · · · · · · · · · · · · · 0.100 3.320 0.140 3.156 0.182 3.018 0.225 2.897 0.269 2.790 0.313 2.695 0.356 2.612 0.398 2.540 0.438 2.475 0.477 2.418 0.514 2.366 0.550 2.321 0.584 2.280 0.616 2.243 0.648 2.210 0.677 2.180 0.705 2.153 0.733 2.128 0.759 2.106 0.784 2.085 0.807 2.066 0.830 2.049 0.852 2.033 0.873 2.019 0.893 2.006 0.912 1.993 0.930 1.981 0.948 1.970 0.966 1.961 0.982 1.951 0.997 1.943 1.013 1.935 1.027 1.927 1.042 1.920 1.055 1.914 1.068 1.908 1.081 1.902 1.139 1.878 1.189 1.861 1.232 1.848 1.269 1.838 1.303 1.830 1.332 1.824 1.359 1.820 1.383 1.816 1.405 1.814 1.425 1.812 1.503 1.809 1.558 1.811 1.600 1.815 1.632 1.819

k = dL · · · · · · · · · 0.087 0.124 0.162 0.202 0.243 0.285 0.325 0.366 0.405 0.442 0.478 0.513 0.547 0.579 0.611 0.640 0.669 0.696 0.722 0.747 0.771 0.794 0.816 0.837 0.857 0.877 0.896 0.914 0.931 0.948 0.964 0.980 0.995 1.009 1.023 1.037 1.099 1.152 1.198 1.238 1.273 1.305 1.333 1.358 1.381 1.403 1.486 1.544 1.588 1.621

10 dU · · · · · · · · · 3.403 3.251 3.119 3.003 2.898 2.804 2.720 2.646 2.581 2.521 2.468 2.419 2.375 2.336 2.302 2.269 2.239 2.212 2.187 2.164 2.143 2.124 2.106 2.089 2.074 2.060 2.046 2.034 2.023 2.012 2.002 1.992 1.984 1.975 1.968 1.960 1.930 1.907 1.890 1.876 1.865 1.857 1.849 1.844 1.840 1.837 1.828 1.826 1.828 1.830

k = dL · · · · · · · · · · 0.077 0.110 0.145 0.182 0.221 0.260 0.299 0.337 0.375 0.411 0.447 0.481 0.514 0.545 0.576 0.606 0.634 0.661 0.687 0.712 0.736 0.759 0.781 0.803 0.824 0.844 0.863 0.881 0.899 0.916 0.932 0.948 0.964 0.979 0.993 1.058 1.115 1.163 1.206 1.243 1.277 1.306 1.334 1.358 1.381 1.468 1.530 1.575 1.611

11 dU · · · · · · · · · · 3.471 3.331 3.206 3.095 2.993 2.900 2.818 2.744 2.677 2.616 2.561 2.511 2.466 2.424 2.386 2.352 2.320 2.291 2.264 2.240 2.216 2.195 2.175 2.157 2.140 2.125 2.110 2.095 2.083 2.071 2.059 2.049 2.039 2.029 2.021 1.982 1.954 1.932 1.914 1.900 1.889 1.879 1.872 1.866 1.862 1.847 1.842 1.841 1.841

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2B.1 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

27

v.1.00

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.025 k = 12 dL 0.068 0.099 0.131 0.165 0.202 0.238 0.275 0.312 0.348 0.383 0.417 0.451 0.483 0.514 0.545 0.573 0.602 0.628 0.655 0.680 0.704 0.727 0.749 0.771 0.792 0.812 0.831 0.850 0.868 0.885 0.902 0.918 0.934 0.949 1.018 1.077 1.129 1.173 1.213 1.249 1.281 1.309 1.335 1.358 1.450 1.515 1.563 1.600

dU 3.529 3.399 3.281 3.175 3.077 2.987 2.905 2.831 2.764 2.703 2.647 2.596 2.549 2.506 2.467 2.430 2.397 2.366 2.338 2.311 2.286 2.264 2.243 2.222 2.204 2.187 2.171 2.155 2.141 2.128 2.115 2.103 2.092 2.082 2.037 2.002 1.975 1.954 1.936 1.922 1.911 1.901 1.893 1.887 1.866 1.857 1.854 1.853

k = dL · 0.061 0.089 0.119 0.150 0.184 0.219 0.254 0.289 0.324 0.358 0.391 0.423 0.455 0.485 0.515 0.544 0.572 0.598 0.624 0.649 0.673 0.696 0.719 0.740 0.761 0.781 0.801 0.820 0.838 0.856 0.873 0.889 0.905 0.978 1.039 1.094 1.141 1.183 1.221 1.254 1.284 1.311 1.336 1.433 1.500 1.550 1.589

13 dU · 3.578 3.457 3.347 3.245 3.151 3.063 2.984 2.911 2.844 2.782 2.726 2.674 2.626 2.582 2.542 2.504 2.470 2.437 2.407 2.379 2.353 2.329 2.307 2.285 2.266 2.247 2.229 2.213 2.198 2.183 2.169 2.156 2.144 2.092 2.051 2.020 1.994 1.973 1.957 1.943 1.931 1.921 1.913 1.886 1.873 1.867 1.864

k = dL · · 0.055 0.080 0.108 0.138 0.169 0.202 0.235 0.269 0.302 0.335 0.367 0.398 0.429 0.459 0.488 0.516 0.543 0.570 0.596 0.621 0.645 0.668 0.690 0.712 0.732 0.753 0.772 0.791 0.810 0.827 0.844 0.861 0.937 1.002 1.059 1.109 1.153 1.192 1.227 1.259 1.287 1.313 1.415 1.485 1.538 1.578

14 dU · · 3.619 3.507 3.404 3.307 3.217 3.133 3.055 2.983 2.917 2.855 2.799 2.747 2.698 2.653 2.612 2.574 2.537 2.504 2.473 2.444 2.417 2.391 2.368 2.345 2.324 2.305 2.286 2.269 2.252 2.236 2.222 2.207 2.148 2.102 2.065 2.035 2.011 1.991 1.975 1.961 1.949 1.939 1.906 1.889 1.880 1.876

k = dL · · · 0.050 0.073 0.098 0.126 0.156 0.187 0.219 0.251 0.282 0.314 0.345 0.375 0.405 0.434 0.463 0.491 0.518 0.544 0.569 0.593 0.617 0.640 0.662 0.684 0.705 0.725 0.745 0.764 0.782 0.801 0.818 0.896 0.965 1.024 1.076 1.122 1.163 1.200 1.233 1.263 1.290 1.397 1.471 1.525 1.567

15 dU · · · 3.655 3.551 3.454 3.362 3.275 3.194 3.119 3.048 2.983 2.922 2.866 2.814 2.765 2.720 2.678 2.639 2.602 2.568 2.536 2.506 2.478 2.451 2.426 2.403 2.381 2.360 2.341 2.322 2.305 2.289 2.273 2.205 2.153 2.111 2.077 2.049 2.026 2.007 1.991 1.977 1.965 1.926 1.905 1.894 1.887

k = dL · · · · 0.045 0.067 0.090 0.116 0.144 0.173 0.203 0.234 0.264 0.295 0.325 0.354 0.383 0.412 0.440 0.467 0.493 0.519 0.544 0.568 0.592 0.615 0.636 0.658 0.679 0.699 0.719 0.738 0.756 0.774 0.856 0.927 0.990 1.044 1.092 1.135 1.173 1.208 1.239 1.268 1.378 1.456 1.512 1.556

16 dU · · · · 3.687 3.589 3.498 3.411 3.328 3.250 3.176 3.108 3.044 2.984 2.928 2.876 2.827 2.782 2.740 2.700 2.662 2.628 2.595 2.564 2.535 2.508 2.482 2.458 2.435 2.414 2.393 2.374 2.355 2.338 2.263 2.204 2.158 2.119 2.088 2.062 2.040 2.022 2.006 1.992 1.946 1.922 1.908 1.899

k = dL · · · · · 0.041 0.061 0.083 0.107 0.134 0.161 0.190 0.219 0.248 0.277 0.306 0.335 0.363 0.391 0.418 0.444 0.470 0.496 0.520 0.544 0.567 0.590 0.612 0.633 0.654 0.675 0.694 0.714 0.731 0.816 0.890 0.954 1.011 1.061 1.106 1.146 1.182 1.215 1.245 1.360 1.441 1.500 1.545

17 dU · · · · · 3.714 3.623 3.537 3.454 3.375 3.300 3.229 3.162 3.100 3.041 2.986 2.934 2.886 2.840 2.797 2.758 2.720 2.685 2.651 2.620 2.590 2.562 2.536 2.511 2.487 2.465 2.443 2.424 2.404 2.322 2.257 2.205 2.163 2.127 2.099 2.074 2.053 2.035 2.020 1.967 1.938 1.921 1.911

k = dL · · · · · · 0.038 0.056 0.077 0.100 0.124 0.151 0.177 0.205 0.233 0.261 0.289 0.317 0.344 0.371 0.398 0.424 0.449 0.474 0.498 0.522 0.545 0.567 0.589 0.610 0.631 0.651 0.671 0.689 0.777 0.853 0.919 0.979 1.031 1.077 1.119 1.157 1.190 1.221 1.342 1.425 1.487 1.534

18 dU · · · · · · 3.737 3.653 3.571 3.494 3.418 3.346 3.277 3.212 3.151 3.093 3.039 2.988 2.940 2.895 2.852 2.812 2.774 2.738 2.704 2.673 2.642 2.614 2.587 2.561 2.537 2.514 2.492 2.472 2.382 2.311 2.253 2.206 2.167 2.135 2.108 2.085 2.065 2.048 1.988 1.955 1.935 1.923

k = dL · · · · · · · 0.035 0.052 0.071 0.092 0.116 0.141 0.167 0.193 0.220 0.247 0.273 0.300 0.327 0.353 0.379 0.404 0.429 0.454 0.477 0.501 0.523 0.545 0.567 0.588 0.608 0.628 0.648 0.738 0.816 0.885 0.946 1.000 1.048 1.092 1.131 1.166 1.198 1.323 1.410 1.474 1.523

19 dU · · · · · · · 3.758 3.679 3.602 3.528 3.456 3.387 3.321 3.258 3.198 3.142 3.089 3.038 2.991 2.945 2.903 2.863 2.824 2.789 2.755 2.723 2.692 2.663 2.636 2.609 2.585 2.561 2.539 2.441 2.364 2.302 2.251 2.208 2.172 2.143 2.116 2.095 2.075 2.009 1.972 1.949 1.934

k = dL · · · · · · · · 0.032 0.048 0.066 0.086 0.108 0.132 0.156 0.181 0.207 0.233 0.259 0.285 0.311 0.336 0.361 0.386 0.410 0.434 0.458 0.481 0.503 0.525 0.546 0.567 0.587 0.607 0.699 0.780 0.851 0.913 0.970 1.019 1.064 1.105 1.141 1.175 1.305 1.395 1.461 1.511

20 dU · · · · · · · · 3.777 3.702 3.630 3.560 3.491 3.425 3.361 3.300 3.242 3.187 3.134 3.085 3.038 2.994 2.951 2.911 2.873 2.837 2.803 2.770 2.740 2.710 2.682 2.656 2.631 2.607 2.502 2.419 2.351 2.295 2.249 2.210 2.177 2.149 2.125 2.104 2.031 1.989 1.964 1.947

k = dL · · · · · · · · · · · · · 0.022 0.034 0.047 0.062 0.079 0.098 0.117 0.137 0.158 0.179 0.201 0.223 0.245 0.267 0.289 0.311 0.332 0.354 0.375 0.396 0.416 0.514 0.603 0.682 0.753 0.818 0.876 0.928 0.975 1.018 1.058 1.211 1.317 1.395 1.454

25 dU · · · · · · · · · · · · · 3.844 3.789 3.734 3.679 3.624 3.570 3.517 3.466 3.416 3.367 3.321 3.276 3.233 3.192 3.152 3.113 3.076 3.042 3.008 2.975 2.944 2.807 2.694 2.601 2.524 2.459 2.404 2.357 2.316 2.281 2.250 2.141 2.077 2.037 2.009

k = dL · · · · · · · · · · · · · · · · · · 0.016 0.025 0.035 0.047 0.061 0.075 0.091 0.107 0.124 0.142 0.160 0.179 0.198 0.217 0.236 0.255 0.350 0.440 0.524 0.600 0.671 0.735 0.793 0.847 0.896 0.940 1.116 1.238 1.328 1.396

30 dU · · · · · · · · · · · · · · · · · · 3.885 3.843 3.800 3.756 3.712 3.667 3.623 3.580 3.537 3.495 3.454 3.414 3.376 3.338 3.302 3.266 3.106 2.970 2.856 2.758 2.676 2.605 2.543 2.490 2.444 2.403 2.257 2.169 2.113 2.074

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2B.2 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

28

v.1.00

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.05 k = 1 dL dU 0.611 1.401 0.699 1.356 0.763 1.332 0.824 1.320 0.879 1.319 0.927 1.324 0.971 1.331 1.010 1.340 1.045 1.351 1.077 1.361 1.106 1.371 1.133 1.382 1.158 1.391 1.180 1.401 1.202 1.411 1.221 1.420 1.239 1.429 1.256 1.437 1.272 1.446 1.287 1.453 1.302 1.461 1.316 1.469 1.329 1.476 1.341 1.483 1.352 1.489 1.363 1.496 1.373 1.502 1.383 1.508 1.393 1.514 1.402 1.519 1.411 1.525 1.419 1.530 1.427 1.535 1.435 1.540 1.442 1.544 1.450 1.549 1.456 1.554 1.463 1.558 1.469 1.562 1.475 1.566 1.481 1.570 1.487 1.574 1.493 1.578 1.498 1.581 1.503 1.585 1.527 1.601 1.549 1.616 1.567 1.630 1.583 1.641 1.598 1.652 1.611 1.662 1.624 1.671 1.634 1.679 1.644 1.687 1.654 1.694 1.692 1.724 1.720 1.746 1.741 1.764 1.758 1.778

k = 2 dL dU · · 0.467 1.896 0.559 1.777 0.629 1.699 0.697 1.641 0.758 1.604 0.812 1.579 0.861 1.562 0.905 1.550 0.945 1.543 0.982 1.539 1.016 1.536 1.046 1.536 1.074 1.535 1.100 1.537 1.125 1.539 1.147 1.541 1.168 1.543 1.188 1.546 1.206 1.549 1.223 1.552 1.240 1.556 1.255 1.560 1.270 1.563 1.284 1.567 1.297 1.570 1.309 1.574 1.321 1.577 1.333 1.581 1.343 1.584 1.354 1.587 1.364 1.591 1.373 1.594 1.382 1.597 1.391 1.600 1.399 1.603 1.407 1.606 1.415 1.609 1.423 1.612 1.430 1.615 1.437 1.617 1.444 1.620 1.450 1.623 1.457 1.626 1.462 1.628 1.490 1.641 1.515 1.652 1.536 1.662 1.554 1.671 1.571 1.680 1.586 1.688 1.600 1.696 1.612 1.703 1.623 1.709 1.634 1.715 1.676 1.741 1.706 1.760 1.730 1.776 1.748 1.789

k = 3 dL dU · · · · 0.367 2.286 0.455 2.128 0.525 2.017 0.595 1.928 0.658 1.864 0.715 1.816 0.767 1.779 0.814 1.750 0.857 1.728 0.897 1.710 0.933 1.696 0.967 1.686 0.997 1.676 1.026 1.669 1.054 1.665 1.078 1.660 1.101 1.656 1.123 1.654 1.143 1.652 1.162 1.651 1.180 1.650 1.197 1.650 1.214 1.650 1.229 1.650 1.244 1.650 1.258 1.651 1.271 1.652 1.283 1.653 1.296 1.654 1.307 1.655 1.318 1.656 1.328 1.658 1.338 1.659 1.348 1.660 1.357 1.662 1.366 1.663 1.375 1.665 1.383 1.666 1.391 1.668 1.399 1.669 1.406 1.671 1.414 1.672 1.421 1.674 1.452 1.681 1.480 1.689 1.504 1.696 1.525 1.703 1.543 1.709 1.560 1.715 1.575 1.721 1.589 1.726 1.602 1.732 1.613 1.736 1.659 1.758 1.693 1.774 1.718 1.788 1.738 1.799

k = 4 dL dU · · · · · · 0.295 2.588 0.376 2.414 0.444 2.283 0.512 2.176 0.575 2.094 0.632 2.030 0.685 1.977 0.734 1.935 0.779 1.901 0.820 1.872 0.859 1.848 0.894 1.828 0.927 1.812 0.957 1.797 0.986 1.786 1.013 1.775 1.038 1.767 1.062 1.759 1.084 1.753 1.105 1.747 1.124 1.742 1.142 1.738 1.160 1.735 1.177 1.732 1.193 1.730 1.208 1.728 1.222 1.726 1.236 1.725 1.249 1.723 1.262 1.722 1.274 1.722 1.285 1.721 1.296 1.721 1.306 1.720 1.316 1.720 1.326 1.720 1.336 1.720 1.345 1.720 1.353 1.720 1.362 1.721 1.370 1.721 1.378 1.721 1.414 1.724 1.444 1.727 1.471 1.731 1.494 1.735 1.515 1.739 1.534 1.743 1.550 1.747 1.566 1.751 1.579 1.754 1.592 1.758 1.643 1.774 1.679 1.788 1.706 1.800 1.728 1.809

k = 5 dL dU · · · · · · · · 0.243 2.821 0.316 2.645 0.379 2.506 0.444 2.389 0.505 2.296 0.562 2.220 0.615 2.157 0.664 2.104 0.710 2.060 0.752 2.022 0.792 1.991 0.829 1.964 0.863 1.940 0.895 1.920 0.925 1.902 0.953 1.886 0.979 1.873 1.004 1.861 1.028 1.850 1.050 1.841 1.071 1.833 1.090 1.825 1.109 1.819 1.127 1.813 1.144 1.808 1.160 1.803 1.175 1.799 1.190 1.795 1.204 1.792 1.217 1.789 1.231 1.786 1.243 1.784 1.254 1.781 1.266 1.779 1.277 1.778 1.287 1.776 1.297 1.775 1.308 1.774 1.317 1.773 1.326 1.772 1.335 1.771 1.374 1.768 1.409 1.767 1.438 1.767 1.464 1.768 1.487 1.770 1.507 1.771 1.525 1.774 1.542 1.776 1.557 1.778 1.571 1.780 1.626 1.792 1.665 1.802 1.694 1.812 1.718 1.820

k = 6 dL dU · · · · · · · · · · 0.203 3.005 0.268 2.832 0.328 2.692 0.389 2.571 0.447 2.472 0.502 2.388 0.554 2.317 0.603 2.258 0.649 2.206 0.692 2.162 0.732 2.124 0.769 2.090 0.804 2.061 0.837 2.035 0.868 2.013 0.897 1.993 0.925 1.975 0.950 1.958 0.975 1.944 0.998 1.931 1.020 1.920 1.041 1.909 1.061 1.900 1.079 1.891 1.098 1.884 1.114 1.876 1.131 1.870 1.146 1.864 1.161 1.859 1.175 1.853 1.189 1.849 1.202 1.845 1.215 1.841 1.227 1.838 1.239 1.835 1.249 1.832 1.261 1.829 1.271 1.826 1.281 1.824 1.291 1.822 1.334 1.813 1.372 1.808 1.404 1.805 1.433 1.802 1.458 1.801 1.480 1.801 1.500 1.801 1.518 1.801 1.535 1.802 1.550 1.803 1.609 1.810 1.651 1.817 1.682 1.824 1.707 1.831

k = 7 dL dU · · · · · · · · · · · · 0.172 3.149 0.230 2.985 0.285 2.847 0.343 2.727 0.398 2.624 0.451 2.536 0.502 2.461 0.549 2.396 0.594 2.339 0.637 2.290 0.677 2.246 0.715 2.208 0.751 2.174 0.784 2.144 0.816 2.118 0.845 2.093 0.874 2.071 0.901 2.052 0.926 2.035 0.949 2.018 0.972 2.004 0.994 1.991 1.015 1.978 1.034 1.968 1.053 1.957 1.071 1.948 1.088 1.939 1.104 1.932 1.120 1.924 1.135 1.918 1.149 1.911 1.163 1.906 1.176 1.900 1.189 1.895 1.202 1.891 1.213 1.887 1.225 1.882 1.235 1.879 1.246 1.875 1.294 1.861 1.335 1.850 1.370 1.843 1.401 1.837 1.428 1.834 1.453 1.831 1.474 1.829 1.494 1.827 1.512 1.827 1.528 1.826 1.592 1.828 1.637 1.832 1.670 1.837 1.697 1.841

k = 8 dL dU · · · · · · · · · · · · · · 0.147 3.266 0.200 3.111 0.251 2.979 0.304 2.860 0.357 2.757 0.407 2.667 0.456 2.589 0.502 2.521 0.546 2.461 0.588 2.407 0.628 2.360 0.666 2.318 0.702 2.280 0.736 2.246 0.767 2.216 0.798 2.189 0.827 2.164 0.853 2.141 0.879 2.120 0.904 2.102 0.928 2.085 0.950 2.069 0.971 2.054 0.991 2.041 1.011 2.029 1.029 2.017 1.047 2.007 1.064 1.997 1.080 1.988 1.096 1.980 1.111 1.972 1.125 1.965 1.139 1.958 1.153 1.952 1.165 1.945 1.178 1.940 1.190 1.935 1.201 1.930 1.253 1.909 1.297 1.894 1.336 1.883 1.369 1.873 1.399 1.867 1.425 1.861 1.448 1.857 1.469 1.854 1.489 1.852 1.506 1.850 1.574 1.846 1.622 1.846 1.658 1.849 1.686 1.852

k = 9 dL dU · · · · · · · · · · · · · · · · 0.127 3.360 0.175 3.216 0.222 3.089 0.272 2.974 0.321 2.873 0.369 2.783 0.416 2.704 0.461 2.634 0.504 2.570 0.545 2.515 0.584 2.464 0.621 2.419 0.657 2.379 0.690 2.342 0.722 2.308 0.753 2.278 0.782 2.251 0.810 2.226 0.836 2.203 0.861 2.182 0.885 2.162 0.908 2.144 0.930 2.127 0.951 2.112 0.971 2.098 0.990 2.085 1.008 2.072 1.025 2.061 1.042 2.050 1.058 2.040 1.074 2.031 1.089 2.022 1.103 2.014 1.117 2.006 1.130 1.999 1.143 1.992 1.155 1.986 1.212 1.959 1.260 1.939 1.301 1.923 1.337 1.910 1.369 1.901 1.397 1.893 1.422 1.887 1.445 1.881 1.465 1.877 1.484 1.874 1.557 1.864 1.608 1.861 1.646 1.862 1.675 1.863

k = dL · · · · · · · · · 0.111 0.155 0.198 0.244 0.290 0.336 0.380 0.424 0.466 0.506 0.544 0.581 0.616 0.649 0.682 0.712 0.741 0.769 0.796 0.821 0.845 0.868 0.891 0.912 0.932 0.952 0.971 0.988 1.006 1.022 1.039 1.054 1.069 1.083 1.097 1.110 1.171 1.222 1.266 1.305 1.339 1.369 1.395 1.420 1.442 1.462 1.539 1.594 1.634 1.665

10 dU · · · · · · · · · 3.438 3.304 3.184 3.073 2.974 2.885 2.806 2.734 2.670 2.613 2.560 2.513 2.470 2.431 2.396 2.363 2.333 2.306 2.280 2.257 2.236 2.216 2.198 2.180 2.164 2.149 2.136 2.123 2.111 2.099 2.088 2.078 2.069 2.060 2.052 2.044 2.010 1.984 1.964 1.948 1.935 1.925 1.916 1.909 1.903 1.898 1.883 1.877 1.875 1.874

k = dL · · · · · · · · · · 0.098 0.138 0.177 0.220 0.264 0.307 0.349 0.391 0.431 0.470 0.508 0.544 0.579 0.612 0.643 0.674 0.703 0.731 0.758 0.783 0.808 0.831 0.853 0.875 0.896 0.916 0.935 0.953 0.971 0.988 1.004 1.020 1.036 1.050 1.065 1.129 1.183 1.231 1.272 1.308 1.340 1.369 1.394 1.418 1.439 1.522 1.579 1.621 1.654

11 dU · · · · · · · · · · 3.503 3.378 3.265 3.159 3.063 2.976 2.897 2.826 2.761 2.703 2.649 2.600 2.556 2.514 2.477 2.443 2.411 2.382 2.355 2.330 2.306 2.285 2.264 2.246 2.229 2.212 2.197 2.182 2.169 2.156 2.144 2.133 2.122 2.112 2.103 2.062 2.031 2.006 1.986 1.971 1.957 1.946 1.937 1.929 1.923 1.902 1.892 1.888 1.886

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2C.1 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

29

v.1.00

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.05 k = 12 dL dU 0.087 3.557 0.123 3.442 0.160 3.335 0.200 3.234 0.240 3.141 0.281 3.057 0.322 2.979 0.362 2.908 0.401 2.844 0.438 2.785 0.475 2.730 0.510 2.680 0.544 2.635 0.577 2.592 0.608 2.554 0.638 2.517 0.668 2.484 0.695 2.453 0.722 2.425 0.748 2.399 0.772 2.374 0.796 2.351 0.819 2.329 0.840 2.309 0.861 2.290 0.881 2.272 0.901 2.256 0.919 2.240 0.938 2.225 0.955 2.212 0.972 2.198 0.988 2.186 1.003 2.174 1.019 2.163 1.087 2.116 1.145 2.079 1.195 2.049 1.239 2.025 1.277 2.006 1.311 1.990 1.342 1.977 1.369 1.966 1.394 1.956 1.416 1.948 1.504 1.922 1.564 1.908 1.609 1.901 1.643 1.897

k = dL · 0.078 0.111 0.145 0.182 0.220 0.259 0.297 0.336 0.373 0.409 0.445 0.479 0.513 0.545 0.576 0.605 0.634 0.662 0.688 0.714 0.739 0.762 0.785 0.807 0.829 0.849 0.869 0.888 0.906 0.923 0.941 0.957 0.973 1.045 1.106 1.160 1.206 1.246 1.283 1.315 1.344 1.370 1.393 1.486 1.549 1.596 1.632

13 dU · 3.603 3.496 3.396 3.300 3.211 3.128 3.052 2.983 2.919 2.860 2.805 2.754 2.708 2.665 2.625 2.588 2.554 2.522 2.492 2.464 2.438 2.413 2.391 2.369 2.349 2.330 2.312 2.296 2.280 2.265 2.251 2.237 2.224 2.170 2.127 2.094 2.066 2.043 2.024 2.009 1.995 1.984 1.974 1.941 1.924 1.914 1.908

k = dL · · 0.070 0.100 0.132 0.166 0.202 0.239 0.275 0.312 0.348 0.383 0.417 0.451 0.483 0.515 0.545 0.575 0.604 0.631 0.657 0.683 0.707 0.731 0.754 0.776 0.797 0.818 0.838 0.857 0.875 0.893 0.910 0.927 1.003 1.068 1.124 1.172 1.215 1.254 1.288 1.318 1.345 1.371 1.467 1.534 1.583 1.621

14 dU · · 3.642 3.543 3.448 3.358 3.272 3.193 3.119 3.050 2.987 2.928 2.874 2.823 2.776 2.733 2.692 2.654 2.619 2.585 2.555 2.526 2.499 2.473 2.449 2.427 2.406 2.386 2.367 2.349 2.333 2.316 2.301 2.287 2.225 2.177 2.138 2.106 2.080 2.059 2.040 2.025 2.011 2.000 1.961 1.940 1.927 1.920

k = dL · · · 0.063 0.091 0.120 0.153 0.186 0.221 0.256 0.291 0.325 0.359 0.392 0.425 0.457 0.488 0.518 0.547 0.575 0.602 0.628 0.653 0.678 0.701 0.724 0.747 0.768 0.788 0.808 0.827 0.846 0.864 0.882 0.961 1.029 1.088 1.139 1.184 1.224 1.260 1.292 1.321 1.347 1.449 1.519 1.571 1.610

15 dU · · · 3.676 3.583 3.495 3.409 3.327 3.251 3.179 3.112 3.050 2.992 2.937 2.887 2.840 2.796 2.754 2.716 2.680 2.646 2.615 2.585 2.557 2.530 2.505 2.482 2.460 2.439 2.419 2.401 2.383 2.367 2.351 2.281 2.227 2.183 2.147 2.118 2.094 2.073 2.055 2.040 2.026 1.981 1.956 1.941 1.931

k = dL · · · · 0.058 0.083 0.110 0.141 0.172 0.205 0.238 0.272 0.305 0.337 0.370 0.401 0.432 0.462 0.492 0.520 0.548 0.575 0.601 0.626 0.650 0.673 0.696 0.718 0.739 0.760 0.780 0.799 0.818 0.836 0.919 0.990 1.052 1.105 1.153 1.195 1.232 1.266 1.296 1.324 1.431 1.504 1.558 1.599

16 dU · · · · 3.706 3.619 3.535 3.454 3.376 3.302 3.233 3.168 3.107 3.050 2.996 2.946 2.899 2.854 2.813 2.774 2.738 2.703 2.671 2.641 2.612 2.585 2.559 2.535 2.512 2.491 2.470 2.451 2.432 2.414 2.338 2.278 2.230 2.190 2.156 2.129 2.105 2.086 2.068 2.053 2.002 1.972 1.955 1.943

k = dL · · · · · 0.052 0.076 0.101 0.130 0.160 0.191 0.222 0.254 0.286 0.318 0.349 0.379 0.410 0.439 0.467 0.495 0.523 0.549 0.574 0.599 0.623 0.647 0.669 0.691 0.713 0.733 0.753 0.773 0.791 0.877 0.951 1.016 1.072 1.121 1.165 1.205 1.240 1.272 1.301 1.412 1.489 1.545 1.588

17 dU · · · · · 3.731 3.650 3.572 3.494 3.420 3.349 3.282 3.219 3.160 3.103 3.050 3.000 2.954 2.909 2.868 2.829 2.792 2.757 2.724 2.694 2.664 2.637 2.610 2.586 2.562 2.540 2.518 2.499 2.479 2.396 2.330 2.276 2.232 2.195 2.165 2.139 2.116 2.097 2.080 2.022 1.989 1.968 1.955

k = dL · · · · · · 0.048 0.070 0.094 0.120 0.149 0.178 0.208 0.238 0.269 0.299 0.329 0.359 0.388 0.417 0.445 0.472 0.499 0.525 0.550 0.575 0.599 0.622 0.644 0.666 0.688 0.708 0.728 0.747 0.836 0.913 0.980 1.038 1.090 1.136 1.177 1.214 1.247 1.277 1.394 1.474 1.532 1.577

18 dU · · · · · · 3.753 3.678 3.604 3.531 3.460 3.392 3.327 3.266 3.208 3.153 3.101 3.051 3.005 2.961 2.919 2.880 2.843 2.808 2.775 2.744 2.714 2.686 2.660 2.634 2.610 2.587 2.565 2.544 2.454 2.382 2.323 2.275 2.235 2.201 2.172 2.148 2.126 2.108 2.043 2.006 1.982 1.967

k = dL · · · · · · · 0.044 0.065 0.087 0.112 0.138 0.166 0.195 0.224 0.253 0.282 0.312 0.340 0.369 0.396 0.424 0.451 0.477 0.502 0.527 0.551 0.575 0.598 0.620 0.642 0.663 0.684 0.703 0.795 0.874 0.944 1.005 1.058 1.106 1.149 1.187 1.222 1.253 1.375 1.458 1.519 1.565

19 dU · · · · · · · 3.773 3.702 3.633 3.563 3.495 3.430 3.368 3.309 3.252 3.198 3.147 3.099 3.053 3.009 2.968 2.929 2.891 2.857 2.823 2.792 2.762 2.733 2.706 2.680 2.655 2.632 2.610 2.512 2.435 2.371 2.318 2.275 2.238 2.206 2.179 2.156 2.135 2.064 2.022 1.996 1.979

k = dL · · · · · · · · 0.041 0.060 0.081 0.104 0.129 0.156 0.183 0.211 0.239 0.267 0.295 0.323 0.351 0.378 0.404 0.430 0.456 0.481 0.506 0.530 0.553 0.576 0.598 0.619 0.640 0.660 0.754 0.836 0.908 0.971 1.027 1.076 1.121 1.161 1.197 1.229 1.356 1.443 1.506 1.554

20 dU · · · · · · · · 3.790 3.724 3.658 3.592 3.528 3.465 3.406 3.348 3.293 3.240 3.190 3.142 3.097 3.054 3.013 2.974 2.937 2.902 2.869 2.837 2.807 2.778 2.751 2.724 2.699 2.676 2.572 2.488 2.419 2.362 2.315 2.275 2.241 2.211 2.186 2.164 2.085 2.040 2.010 1.991

k = dL · · · · · · · · · · · · · 0.028 0.042 0.058 0.076 0.095 0.115 0.137 0.159 0.182 0.206 0.229 0.253 0.277 0.300 0.324 0.347 0.370 0.392 0.415 0.437 0.458 0.560 0.651 0.732 0.805 0.870 0.928 0.980 1.027 1.070 1.109 1.260 1.364 1.439 1.496

25 dU · · · · · · · · · · · · · 3.854 3.804 3.754 3.703 3.652 3.601 3.551 3.503 3.455 3.409 3.365 3.322 3.281 3.241 3.203 3.166 3.130 3.096 3.063 3.032 3.002 2.867 2.756 2.664 2.586 2.521 2.465 2.418 2.376 2.340 2.308 2.195 2.127 2.083 2.053

k = dL · · · · · · · · · · · · · · · · · · 0.021 0.031 0.043 0.057 0.072 0.089 0.106 0.125 0.144 0.163 0.183 0.203 0.224 0.244 0.264 0.285 0.385 0.479 0.566 0.645 0.717 0.782 0.841 0.895 0.944 0.989 1.163 1.284 1.371 1.437

30 dU · · · · · · · · · · · · · · · · · · 3.892 3.854 3.815 3.774 3.733 3.691 3.650 3.609 3.569 3.529 3.490 3.452 3.415 3.379 3.344 3.310 3.155 3.023 2.910 2.814 2.732 2.662 2.600 2.547 2.500 2.458 2.309 2.218 2.159 2.117

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2C.2 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

30

v.1.00

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.10 k = 1 dL dU 0.789 1.612 0.866 1.567 0.935 1.530 0.997 1.516 1.050 1.511 1.096 1.510 1.137 1.513 1.172 1.516 1.205 1.522 1.234 1.528 1.260 1.534 1.284 1.541 1.306 1.547 1.326 1.554 1.345 1.560 1.362 1.566 1.378 1.573 1.393 1.578 1.406 1.584 1.420 1.589 1.432 1.595 1.444 1.600 1.455 1.606 1.465 1.610 1.475 1.615 1.484 1.619 1.492 1.624 1.501 1.628 1.509 1.632 1.517 1.636 1.524 1.640 1.531 1.644 1.538 1.648 1.544 1.651 1.551 1.654 1.557 1.658 1.562 1.661 1.568 1.664 1.573 1.667 1.578 1.670 1.583 1.673 1.588 1.676 1.593 1.679 1.597 1.681 1.602 1.684 1.622 1.697 1.639 1.708 1.654 1.717 1.668 1.726 1.680 1.734 1.690 1.741 1.700 1.748 1.709 1.754 1.717 1.760 1.725 1.766 1.756 1.788 1.778 1.805 1.795 1.819 1.809 1.829

k = 2 dL dU · · 0.608 2.078 0.694 1.970 0.775 1.881 0.846 1.825 0.908 1.786 0.963 1.757 1.011 1.737 1.053 1.721 1.093 1.710 1.127 1.702 1.159 1.696 1.188 1.691 1.214 1.688 1.238 1.686 1.261 1.685 1.281 1.685 1.300 1.685 1.318 1.685 1.335 1.686 1.350 1.687 1.365 1.688 1.379 1.690 1.392 1.691 1.404 1.693 1.415 1.694 1.426 1.696 1.437 1.698 1.447 1.700 1.456 1.701 1.465 1.703 1.474 1.705 1.482 1.707 1.491 1.709 1.498 1.711 1.505 1.712 1.512 1.714 1.519 1.716 1.526 1.718 1.532 1.719 1.538 1.721 1.544 1.723 1.549 1.725 1.555 1.726 1.560 1.728 1.584 1.736 1.604 1.743 1.622 1.750 1.638 1.756 1.652 1.762 1.665 1.768 1.676 1.773 1.687 1.778 1.696 1.782 1.705 1.787 1.739 1.805 1.765 1.819 1.784 1.830 1.799 1.840

k = 3 dL dU · · · · 0.479 2.439 0.565 2.298 0.649 2.181 0.724 2.097 0.789 2.033 0.849 1.983 0.902 1.944 0.949 1.912 0.992 1.887 1.031 1.867 1.066 1.850 1.099 1.836 1.128 1.824 1.156 1.815 1.182 1.808 1.205 1.800 1.227 1.795 1.247 1.790 1.266 1.786 1.284 1.783 1.300 1.780 1.316 1.778 1.331 1.776 1.345 1.774 1.359 1.773 1.371 1.772 1.383 1.771 1.395 1.770 1.405 1.770 1.416 1.770 1.426 1.770 1.435 1.770 1.444 1.770 1.453 1.770 1.461 1.770 1.469 1.770 1.477 1.771 1.484 1.771 1.491 1.771 1.498 1.772 1.504 1.772 1.511 1.773 1.517 1.774 1.545 1.777 1.569 1.780 1.590 1.784 1.608 1.788 1.624 1.791 1.639 1.795 1.651 1.798 1.663 1.802 1.674 1.805 1.684 1.808 1.723 1.822 1.751 1.833 1.772 1.842 1.789 1.850

k = 4 dL dU · · · · · · 0.387 2.716 0.468 2.562 0.550 2.431 0.624 2.329 0.691 2.250 0.752 2.185 0.807 2.132 0.857 2.088 0.903 2.052 0.945 2.022 0.983 1.996 1.018 1.974 1.050 1.955 1.080 1.938 1.108 1.924 1.134 1.912 1.158 1.901 1.180 1.892 1.201 1.884 1.221 1.876 1.239 1.870 1.257 1.864 1.273 1.859 1.289 1.854 1.304 1.850 1.318 1.846 1.331 1.843 1.344 1.840 1.356 1.838 1.367 1.835 1.378 1.833 1.389 1.832 1.399 1.830 1.408 1.828 1.418 1.827 1.427 1.826 1.435 1.825 1.444 1.824 1.451 1.823 1.459 1.822 1.466 1.822 1.473 1.821 1.505 1.819 1.533 1.819 1.556 1.819 1.577 1.820 1.595 1.821 1.612 1.823 1.626 1.824 1.640 1.826 1.652 1.828 1.663 1.830 1.706 1.839 1.737 1.847 1.760 1.854 1.779 1.860

k = 5 dL dU · · · · · · · · 0.318 2.930 0.393 2.774 0.471 2.637 0.542 2.528 0.608 2.439 0.669 2.364 0.725 2.301 0.776 2.248 0.824 2.204 0.867 2.165 0.907 2.132 0.944 2.103 0.978 2.078 1.010 2.055 1.040 2.036 1.067 2.019 1.093 2.003 1.117 1.990 1.140 1.978 1.161 1.966 1.181 1.957 1.200 1.948 1.218 1.940 1.235 1.932 1.251 1.926 1.267 1.920 1.281 1.914 1.295 1.909 1.308 1.904 1.320 1.900 1.333 1.896 1.344 1.893 1.355 1.889 1.365 1.886 1.376 1.883 1.385 1.880 1.395 1.878 1.404 1.876 1.412 1.874 1.421 1.872 1.429 1.870 1.465 1.863 1.496 1.859 1.523 1.855 1.546 1.853 1.567 1.852 1.585 1.851 1.601 1.851 1.616 1.851 1.629 1.851 1.641 1.852 1.689 1.856 1.723 1.861 1.748 1.866 1.768 1.871

k = 6 dL dU · · · · · · · · · · 0.265 3.097 0.335 2.945 0.407 2.809 0.475 2.696 0.539 2.602 0.599 2.521 0.654 2.452 0.706 2.393 0.753 2.341 0.798 2.297 0.839 2.258 0.877 2.224 0.913 2.193 0.946 2.166 0.977 2.142 1.006 2.121 1.033 2.101 1.058 2.084 1.082 2.068 1.105 2.054 1.126 2.041 1.147 2.029 1.166 2.018 1.184 2.008 1.201 1.999 1.217 1.991 1.233 1.983 1.248 1.976 1.262 1.969 1.275 1.963 1.288 1.958 1.301 1.952 1.313 1.948 1.324 1.943 1.335 1.939 1.345 1.935 1.356 1.931 1.365 1.928 1.374 1.924 1.384 1.921 1.424 1.909 1.459 1.899 1.488 1.893 1.514 1.887 1.537 1.884 1.557 1.881 1.575 1.878 1.591 1.877 1.606 1.876 1.620 1.875 1.672 1.874 1.709 1.876 1.736 1.878 1.758 1.882

k = 7 dL dU · · · · · · · · · · · · 0.225 3.229 0.288 3.084 0.355 2.953 0.419 2.840 0.480 2.743 0.538 2.659 0.593 2.586 0.644 2.522 0.691 2.466 0.736 2.417 0.777 2.374 0.816 2.335 0.852 2.300 0.886 2.270 0.918 2.242 0.948 2.216 0.977 2.194 1.003 2.173 1.028 2.154 1.052 2.137 1.074 2.121 1.095 2.107 1.115 2.094 1.135 2.082 1.153 2.070 1.170 2.060 1.187 2.050 1.203 2.041 1.218 2.033 1.232 2.025 1.246 2.018 1.259 2.011 1.271 2.004 1.284 1.999 1.296 1.993 1.307 1.988 1.317 1.983 1.328 1.978 1.338 1.974 1.383 1.956 1.421 1.942 1.454 1.931 1.482 1.922 1.507 1.916 1.529 1.910 1.549 1.906 1.567 1.903 1.583 1.900 1.598 1.898 1.655 1.892 1.695 1.890 1.724 1.891 1.747 1.892

k = 8 dL dU · · · · · · · · · · · · · · 0.193 3.335 0.250 3.199 0.312 3.073 0.372 2.962 0.430 2.865 0.486 2.780 0.539 2.705 0.589 2.639 0.636 2.580 0.680 2.527 0.721 2.480 0.760 2.438 0.797 2.400 0.832 2.366 0.864 2.335 0.895 2.307 0.924 2.282 0.951 2.258 0.977 2.237 1.002 2.217 1.025 2.199 1.047 2.182 1.068 2.166 1.088 2.152 1.107 2.139 1.125 2.127 1.143 2.115 1.159 2.104 1.175 2.094 1.190 2.085 1.205 2.076 1.218 2.068 1.232 2.060 1.245 2.053 1.257 2.046 1.269 2.040 1.280 2.034 1.291 2.028 1.341 2.003 1.382 1.984 1.419 1.970 1.450 1.958 1.477 1.949 1.501 1.941 1.523 1.935 1.542 1.929 1.560 1.925 1.575 1.921 1.637 1.910 1.680 1.905 1.712 1.903 1.737 1.903

k = 9 dL dU · · · · · · · · · · · · · · · · 0.167 3.421 0.219 3.294 0.276 3.174 0.332 3.067 0.388 2.972 0.441 2.887 0.491 2.811 0.540 2.744 0.586 2.682 0.630 2.628 0.671 2.578 0.710 2.534 0.747 2.494 0.782 2.457 0.814 2.423 0.846 2.392 0.875 2.364 0.903 2.339 0.930 2.315 0.955 2.293 0.979 2.273 1.002 2.254 1.023 2.237 1.044 2.220 1.064 2.205 1.082 2.191 1.100 2.178 1.118 2.166 1.134 2.155 1.150 2.144 1.165 2.133 1.180 2.124 1.193 2.115 1.207 2.106 1.220 2.098 1.232 2.091 1.244 2.083 1.298 2.053 1.344 2.029 1.383 2.010 1.417 1.995 1.446 1.982 1.473 1.972 1.496 1.964 1.517 1.957 1.536 1.950 1.553 1.945 1.620 1.929 1.666 1.920 1.700 1.916 1.726 1.914

k = dL · · · · · · · · · 0.146 0.194 0.246 0.299 0.351 0.401 0.450 0.497 0.542 0.585 0.626 0.664 0.701 0.735 0.769 0.800 0.830 0.858 0.885 0.911 0.935 0.958 0.981 1.002 1.022 1.042 1.060 1.078 1.095 1.111 1.127 1.142 1.157 1.170 1.184 1.197 1.255 1.305 1.347 1.384 1.416 1.444 1.469 1.492 1.512 1.530 1.602 1.651 1.687 1.715

10 dU · · · · · · · · · 3.491 3.374 3.260 3.157 3.065 2.981 2.906 2.838 2.776 2.720 2.669 2.622 2.579 2.541 2.506 2.473 2.443 2.415 2.389 2.365 2.343 2.323 2.304 2.286 2.269 2.253 2.239 2.225 2.212 2.200 2.189 2.178 2.168 2.158 2.149 2.140 2.103 2.074 2.051 2.032 2.017 2.004 1.993 1.984 1.976 1.970 1.947 1.936 1.929 1.925

k = dL · · · · · · · · · · 0.129 0.172 0.221 0.270 0.319 0.367 0.413 0.459 0.503 0.544 0.584 0.622 0.659 0.693 0.726 0.757 0.787 0.816 0.843 0.869 0.894 0.917 0.940 0.962 0.983 1.003 1.022 1.040 1.057 1.074 1.091 1.106 1.121 1.136 1.149 1.212 1.265 1.310 1.350 1.384 1.415 1.441 1.466 1.488 1.507 1.584 1.636 1.675 1.704

11 dU · · · · · · · · · · 3.550 3.441 3.334 3.236 3.147 3.065 2.990 2.922 2.860 2.803 2.751 2.704 2.660 2.619 2.582 2.548 2.516 2.487 2.460 2.434 2.411 2.388 2.368 2.348 2.330 2.314 2.297 2.282 2.268 2.255 2.242 2.230 2.219 2.208 2.198 2.154 2.120 2.093 2.070 2.052 2.036 2.023 2.012 2.002 1.994 1.966 1.951 1.942 1.937

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2D.1 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

31

v.1.00

n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 125 150 175 200

α = 0.10 k = 12 dL dU 0.114 3.599 0.154 3.498 0.199 3.398 0.245 3.304 0.291 3.219 0.337 3.139 0.381 3.066 0.425 2.999 0.467 2.936 0.507 2.880 0.546 2.827 0.584 2.779 0.620 2.734 0.654 2.692 0.687 2.654 0.718 2.618 0.748 2.586 0.776 2.555 0.804 2.526 0.830 2.499 0.855 2.474 0.879 2.451 0.902 2.429 0.924 2.409 0.945 2.389 0.965 2.371 0.985 2.354 1.003 2.337 1.021 2.322 1.039 2.308 1.055 2.294 1.071 2.281 1.087 2.269 1.102 2.257 1.169 2.207 1.225 2.167 1.274 2.135 1.316 2.109 1.353 2.087 1.385 2.069 1.414 2.054 1.440 2.041 1.463 2.029 1.484 2.019 1.566 1.986 1.621 1.967 1.662 1.955 1.693 1.948

k = dL · 0.102 0.139 0.181 0.223 0.266 0.310 0.352 0.394 0.435 0.474 0.512 0.549 0.584 0.618 0.650 0.681 0.711 0.740 0.767 0.793 0.818 0.842 0.866 0.888 0.909 0.930 0.950 0.969 0.987 1.005 1.022 1.038 1.054 1.125 1.185 1.237 1.282 1.321 1.356 1.386 1.414 1.438 1.461 1.547 1.606 1.649 1.682

13 dU · 3.641 3.547 3.453 3.364 3.282 3.205 3.134 3.068 3.006 2.949 2.897 2.848 2.802 2.761 2.721 2.685 2.651 2.619 2.589 2.561 2.535 2.511 2.488 2.466 2.446 2.426 2.408 2.391 2.375 2.359 2.345 2.331 2.317 2.261 2.215 2.179 2.148 2.123 2.102 2.085 2.069 2.056 2.045 2.005 1.982 1.968 1.959

k = dL · · 0.092 0.126 0.164 0.204 0.245 0.286 0.327 0.366 0.406 0.444 0.481 0.517 0.551 0.584 0.616 0.647 0.677 0.705 0.732 0.759 0.784 0.808 0.832 0.854 0.875 0.896 0.916 0.936 0.954 0.972 0.989 1.006 1.081 1.145 1.200 1.247 1.289 1.326 1.358 1.388 1.414 1.438 1.529 1.591 1.637 1.671

14 dU · · 3.677 3.589 3.501 3.417 3.338 3.264 3.195 3.130 3.070 3.013 2.961 2.912 2.866 2.824 2.784 2.747 2.712 2.679 2.649 2.620 2.593 2.568 2.544 2.521 2.500 2.479 2.460 2.442 2.425 2.408 2.393 2.378 2.314 2.264 2.222 2.188 2.160 2.137 2.116 2.099 2.084 2.071 2.025 1.998 1.982 1.971

k = dL · · · 0.083 0.114 0.150 0.187 0.226 0.265 0.303 0.342 0.379 0.416 0.452 0.487 0.521 0.553 0.585 0.615 0.645 0.673 0.700 0.726 0.751 0.776 0.799 0.821 0.843 0.864 0.884 0.904 0.922 0.941 0.958 1.037 1.105 1.163 1.213 1.257 1.296 1.330 1.361 1.389 1.414 1.510 1.576 1.624 1.660

15 dU · · · 3.707 3.626 3.543 3.464 3.388 3.317 3.250 3.187 3.127 3.072 3.020 2.971 2.926 2.883 2.843 2.805 2.770 2.737 2.705 2.676 2.648 2.622 2.597 2.574 2.551 2.530 2.510 2.491 2.473 2.457 2.440 2.369 2.313 2.267 2.229 2.198 2.171 2.148 2.129 2.112 2.097 2.045 2.014 1.995 1.982

k = dL · · · · 0.076 0.104 0.138 0.173 0.209 0.245 0.283 0.319 0.356 0.391 0.426 0.460 0.493 0.525 0.556 0.586 0.615 0.642 0.670 0.695 0.721 0.745 0.768 0.791 0.812 0.834 0.854 0.873 0.892 0.911 0.994 1.064 1.125 1.178 1.225 1.266 1.302 1.334 1.363 1.390 1.492 1.561 1.611 1.649

16 dU · · · · 3.734 3.658 3.580 3.505 3.433 3.365 3.300 3.238 3.180 3.126 3.075 3.026 2.981 2.938 2.898 2.860 2.824 2.791 2.759 2.729 2.700 2.673 2.648 2.624 2.601 2.579 2.558 2.539 2.520 2.502 2.425 2.363 2.312 2.270 2.235 2.206 2.180 2.159 2.140 2.123 2.065 2.031 2.009 1.994

k = dL · · · · · 0.069 0.096 0.127 0.159 0.194 0.229 0.264 0.299 0.334 0.368 0.402 0.435 0.467 0.498 0.528 0.558 0.586 0.614 0.641 0.667 0.691 0.716 0.739 0.761 0.783 0.804 0.825 0.845 0.863 0.950 1.024 1.088 1.144 1.192 1.235 1.273 1.307 1.338 1.366 1.473 1.545 1.598 1.637

17 dU · · · · · 3.757 3.686 3.613 3.542 3.473 3.408 3.345 3.285 3.229 3.175 3.125 3.077 3.032 2.989 2.949 2.911 2.875 2.841 2.809 2.779 2.750 2.722 2.696 2.672 2.648 2.626 2.605 2.585 2.565 2.481 2.413 2.358 2.312 2.274 2.241 2.213 2.189 2.168 2.150 2.086 2.047 2.022 2.005

k = dL · · · · · · 0.063 0.088 0.117 0.148 0.180 0.213 0.247 0.280 0.314 0.347 0.380 0.412 0.443 0.473 0.503 0.532 0.560 0.587 0.614 0.639 0.664 0.688 0.711 0.734 0.756 0.777 0.797 0.817 0.907 0.984 1.050 1.109 1.160 1.205 1.245 1.281 1.313 1.342 1.454 1.530 1.585 1.626

18 dU · · · · · · 3.777 3.711 3.642 3.575 3.509 3.446 3.386 3.328 3.273 3.221 3.171 3.124 3.079 3.037 2.997 2.959 2.923 2.889 2.857 2.826 2.797 2.769 2.743 2.718 2.694 2.671 2.649 2.628 2.537 2.464 2.404 2.354 2.312 2.277 2.246 2.220 2.197 2.177 2.106 2.064 2.036 2.018

k = dL · · · · · · · 0.058 0.081 0.108 0.137 0.168 0.199 0.231 0.264 0.296 0.328 0.359 0.390 0.421 0.450 0.480 0.508 0.535 0.562 0.588 0.614 0.638 0.662 0.685 0.707 0.729 0.750 0.771 0.863 0.944 1.013 1.074 1.127 1.174 1.216 1.253 1.287 1.318 1.435 1.514 1.571 1.615

19 dU · · · · · · · 3.795 3.733 3.668 3.604 3.542 3.482 3.423 3.367 3.314 3.263 3.215 3.168 3.124 3.082 3.043 3.005 2.968 2.935 2.902 2.871 2.842 2.814 2.787 2.761 2.737 2.714 2.691 2.594 2.515 2.450 2.397 2.351 2.313 2.280 2.252 2.227 2.205 2.127 2.081 2.050 2.029

k = dL · · · · · · · · 0.053 0.075 0.101 0.128 0.157 0.187 0.217 0.248 0.279 0.310 0.340 0.370 0.400 0.429 0.457 0.485 0.512 0.539 0.564 0.589 0.614 0.637 0.660 0.682 0.704 0.725 0.821 0.904 0.976 1.039 1.094 1.143 1.187 1.226 1.261 1.293 1.415 1.498 1.558 1.603

20 dU · · · · · · · · 3.811 3.752 3.692 3.631 3.571 3.514 3.457 3.404 3.352 3.302 3.254 3.209 3.166 3.125 3.085 3.047 3.012 2.977 2.945 2.914 2.884 2.856 2.829 2.803 2.778 2.755 2.651 2.567 2.497 2.440 2.391 2.349 2.314 2.283 2.256 2.232 2.148 2.098 2.064 2.041

k = dL · · · · · · · · · · · · · 0.037 0.053 0.072 0.093 0.115 0.138 0.163 0.188 0.213 0.239 0.265 0.290 0.316 0.341 0.367 0.391 0.416 0.440 0.463 0.486 0.509 0.615 0.709 0.792 0.866 0.931 0.990 1.042 1.089 1.131 1.170 1.318 1.418 1.490 1.545

25 dU · · · · · · · · · · · · · 3.868 3.824 3.778 3.731 3.684 3.637 3.591 3.546 3.501 3.458 3.416 3.375 3.336 3.298 3.261 3.226 3.192 3.159 3.127 3.097 3.068 2.936 2.827 2.735 2.658 2.592 2.536 2.487 2.445 2.408 2.375 2.257 2.184 2.137 2.103

k = dL · · · · · · · · · · · · · · · · · · 0.027 0.039 0.054 0.070 0.088 0.107 0.126 0.147 0.168 0.190 0.211 0.233 0.255 0.278 0.300 0.322 0.428 0.527 0.617 0.698 0.772 0.838 0.898 0.952 1.001 1.046 1.219 1.337 1.421 1.485

30 dU · · · · · · · · · · · · · · · · · · 3.902 3.869 3.833 3.796 3.758 3.719 3.681 3.643 3.605 3.568 3.531 3.496 3.460 3.426 3.393 3.361 3.211 3.083 2.973 2.879 2.798 2.727 2.666 2.612 2.564 2.522 2.369 2.275 2.211 2.167

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

c E. Barrios y J. A. Garc´ıa

TABLA 2D.2 Cotas cr´ıticas dL(α;n,k) , dU(α;n,k) del estad´ıstico de Durbin-Watson.

32

v.1.00

Probabilidad, Inferencia Estad´ıstica y Econometr´ıa

3. 3.1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70

33

Tabla de N´ umeros Aleatorios 1050 N´ umeros Seudoaleatorios 1 82207 08782 03897 31066 70691 05473 66636 72813 50139 08305 09732 06394 11775 05492 84217 35602 18937 09063 68686 94005 68729 61437 67205 81435 35931 29107 90077 94296 78466 24715 87308 96027 50097 13275 34327 44247 26040 78618 87347 80088 48543 12746 54959 63750 31078 43124 43025 79155 69978 95986 89260 70348 36057 14822 14861 92519 33432 85477 96733 95071 31762 74194 08009 19493 81881 65602 64218 15774 14027 51995

2 32971 88748 78310 29277 36787 12332 94524 65324 20777 08720 10854 20244 46653 36719 03832 93562 31233 72987 29310 63168 09294 46400 52661 01444 83937 34375 02749 76997 24707 47704 60352 08859 07108 51460 31136 78593 85864 40504 99792 22869 65583 72369 24443 64262 98829 64862 74904 22329 33341 86287 01121 95862 56642 56346 05403 01665 78296 95660 11979 31306 50916 44604 73767 59379 47467 02360 33987 49562 23287 52431

3 61821 93209 00843 39356 30040 29330 39277 37927 09743 45241 21679 41751 28843 45201 39880 46721 69786 22942 27737 99225 20553 83356 44940 69273 10333 28918 73330 31616 06113 81971 89109 53132 72610 87908 78592 55878 16778 02604 18053 56309 01521 88929 22348 57639 00648 36209 27636 61951 57891 20945 89550 10251 69166 34764 98439 42921 80872 43894 58208 51364 33590 83439 82915 53859 29274 40533 84448 70491 32715 58043

c E. Barrios y J. A. Garc´ıa

4 07315 02753 36634 68641 86232 91235 13253 59485 09944 56392 23452 02024 57869 34011 90499 58406 27423 73570 41013 32729 60615 93696 65867 83712 78759 02651 41414 37554 49373 80046 15387 31129 27253 30894 39142 43412 71500 57240 68564 44646 16999 07235 87977 90103 58021 29283 84706 18687 24062 28844 86257 85670 42068 01974 63040 70981 91904 51113 47759 90976 22597 85492 35867 04117 30934 54264 44521 11895 90935 62695

5 05123 51060 88682 93508 45147 71281 56450 56667 51075 70316 78626 92900 78131 53980 51196 74254 61189 99178 71361 38332 47547 41961 47850 04884 05037 32807 95824 08270 50945 58200 87582 21942 86750 31955 86501 43229 71092 06697 62756 22752 97872 44319 95615 14477 95043 60822 67750 32898 72470 18291 48235 30137 99118 96148 81557 69109 49754 31450 30551 23021 60078 28450 78324 96045 22320 36303 87606 48996 13858 51833

6 49509 20520 01283 67156 26600 56367 27705 31734 08054 22790 03967 98611 14760 16243 27703 18811 73842 74670 70101 20433 95868 82064 19618 12117 03009 86061 65587 67042 95451 93027 35140 78718 28086 03957 72157 18323 13622 23727 08622 91753 10458 34906 61691 79209 59644 66093 72582 02136 81311 87287 35045 89658 22337 31068 96735 24056 55938 27637 38510 18136 96242 14910 94670 43959 73655 19012 05917 70008 21421 92116

7 90787 95553 18719 48082 06145 92870 40385 65678 08237 49750 32638 93004 05781 96598 33593 08743 95420 86925 87117 80957 27575 74905 34954 82171 45646 07634 40375 32544 58550 44994 53967 43613 39437 32518 63527 89169 92687 63192 15321 08212 39045 78656 65636 75450 46876 00365 36718 43242 20359 06384 11439 21653 58790 28547 23649 61950 98134 75221 65681 89260 55968 81475 20362 20782 72771 03235 19605 39219 54507 87174

8 40307 00304 05062 82129 34248 67289 45780 88267 56341 27856 34731 67680 80610 63744 78317 41168 28386 52416 73214 93792 28445 22200 96868 64567 70365 48297 67842 29185 42748 77322 38850 90474 67703 67123 74013 07109 44602 16315 90718 79369 38358 34954 20590 91035 07478 48681 04183 60520 67782 05109 37364 94004 94750 27859 69837 82158 83517 22127 74183 99351 07185 07563 24641 39648 44895 65292 15402 60029 80184 95566

9 17954 79373 58450 77012 55430 15497 46888 67990 60524 88651 13096 43594 98863 62028 98902 27447 07615 42298 13145 42603 75083 22864 98198 26240 71554 02864 49525 15643 77981 30296 90938 61483 17628 16800 76571 60604 49380 99058 29674 50526 00317 74594 28991 04374 71748 58598 22380 87246 29891 22216 28969 41721 79523 65449 08667 73354 95264 14441 61242 03051 01308 45893 53489 88951 87941 43814 76479 35009 00922 09210

10 39359 25619 03007 73662 73805 91458 01992 38285 11440 20872 89622 64250 25172 69860 82734 70694 05336 05322 54469 04600 57291 27586 38242 77157 02507 68798 63674 06072 78868 82056 54470 00138 06197 83785 87286 10544 20154 04616 82069 42499 60506 96445 11429 16419 35035 63975 96030 43448 66677 29425 16983 47262 43035 46979 13695 94756 30988 82722 14056 04622 01061 88970 06511 53974 85621 94427 73181 72325 41586 65367

11 03509 30230 18043 58886 69535 86862 24363 03004 77426 45309 96556 76547 13139 61257 32279 54801 82037 04194 17946 76042 15750 02415 04384 03304 35697 84678 03170 70927 49284 59420 70389 88732 50731 51182 06946 27785 74640 17027 33948 16318 00099 51104 66437 89987 28377 04758 54652 60134 16188 45199 31708 27860 90875 40955 71573 85887 03039 98994 96089 75100 84467 64756 67402 32391 92089 93421 33173 51324 07130 38960

12 41932 70305 82009 12708 70709 66026 24922 37612 12225 45738 49203 77016 84419 80528 00439 95152 09500 59593 22718 58495 11610 69494 93983 74675 47666 96942 48362 71644 07178 77465 35013 75923 29454 35737 13135 35265 69627 00662 49601 12735 93084 54658 06514 18093 78314 95095 16280 82182 83486 17817 09253 87321 47546 99165 60711 48311 19727 89319 39000 05123 25558 78089 90585 52822 40424 05174 38720 28218 76427 99917

13 43282 21380 48034 29393 07467 36217 39112 68859 83228 63223 16957 86194 50756 66348 28388 52571 63224 64814 04640 60616 63546 69663 78381 27520 63533 07619 40050 71104 37543 08361 44122 29391 39432 45671 90821 46138 56552 07209 97345 13826 33072 32172 81454 36065 44956 11269 48384 66493 31076 42741 22202 20735 79240 61754 35702 51023 41776 00185 78819 32264 92507 45884 35593 31603 80042 16599 83412 83398 12043 35136

14 68591 27406 04251 07233 33263 26543 91854 42143 30083 31353 86919 34316 13884 06244 80217 14835 55901 89117 11577 50111 35273 64862 35914 33186 34112 24987 05777 04424 48830 06568 33961 15449 73065 22024 12435 61232 02323 34427 18781 66726 42085 50619 73091 84075 73710 05001 36434 45007 00575 64886 35854 04350 58578 87738 08571 28261 53424 89949 82582 78105 90732 80353 92179 87594 76452 76135 53840 72611 25585 01280

15 47877 84985 18956 91496 75402 94725 27677 99716 08031 75646 82242 13515 93264 19489 18720 59536 05859 81064 44776 07848 17333 81553 46828 50510 90591 92411 08439 16527 43691 49460 54571 04077 03546 02831 11876 49986 97439 97181 04994 90387 33138 06772 93854 82521 42747 23589 41464 93803 60955 82736 98413 55524 92376 34071 01896 15316 78041 11289 94105 38241 63759 01839 92200 35509 88664 60482 43411 36070 57165 16504

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