Cheat Sheet 35c2d

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Cheat Sheet

Some Theorems Involving Sets: (1) (2) (3) (4) (5) (6) (7)

If A ⊂ B and B ⊂ C , then A ⊂ C

(8) A∪B =B∪A (9) A ∪ (B ∪ C) = (A ∪ B) ∪ C = A ∪ B ∪ C (10) A∩B =B∩A (11) A ∩ (B ∩ C) = (A ∩ B) ∩ C = A ∩ B ∩ C (12a) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (12b) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (13)

A − B = A ∩ B0 If A ⊂ B , then A0 ⊃ B 0 or B 0 ⊂ A0 A ∪ ∅ = A, A ∩ ∅ = ∅ A ∪ U = U, A ∩ U = A (A ∪ B)0 = A0 ∩ B 0 (A ∩ B)0 = A0 ∪ B 0 A = (A ∩ B) ∪ (A ∩ B 0 )

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Fundamental principle of counting: If an operation consists of k steps, of which the rst can be

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Permutations and Combinations:

done in n1 ways, for each of these the second step can be done in n2 ways, for each of the rst two the third step can be done in n3 ways, an so forth, then the whole operation can be done in n1 · n2 · · · nk ways.

n! The number of permutation of n distinct objects taken r at a time is n Pr = (n−r)! for r = 0, 1, 2, · · · , n. The number of permutations of n objects of which n1 are of one kind, n2 are of a second kind, · · · , nk are of a k th type, and n = n1 + n2 + · · · + nk is n Pn1 ,n2 ,··· ,nk = n1 !·nn! . 2 !···nk !

The number of combinations of n distinct objects taken r at a time is 0, 1, 2, · · · , n.

Binomial coecient: (x + y) = P x y . • Some Important Theorems on Probability: Let A n

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S , then

n n r=0 r

n r



=

n! r!(n−r)!

for all r =

n−r r

i

(i = 1, 2, 3, · · · n) be events of the sample space

(1) If A1 ⊂ A2 then P (A1 ) ≤ P (A2 ) (2) For every event A, 0 ≤ P (A) ≤ 1 0 0 (3) If A is the complement of A, then P (A ) = 1 − P (A) (4) P (∅) = 0 (5) If A and B are any two events, then P (A ∪ B) = P (A) + P (B) − P (A ∩ B). More generally, if A1 , A2 , A3 are any three events, then P (A1 ∪ A2 ∪ A3 ) = P (A1 ) + P (A2 ) + P (A3 ) − P (A1 ∩ A2 ) − P (A2 ∩ A3 ) − P (A3 ∩ A1 ) + P (A1 ∩ A2 ∩ A3 ). •

Conditional probability of B given A: P (B|A) =

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Independent Events: Events A and B are independent if and only if P (A ∩ B) = P (A)P (B), or in

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or P (A ∩ B) = P (A)P (B|A). For any three events A1 , A2 , A3 , we have P (A1 ∩ A2 ∩ A3 ) = P (A1 )P (A2 |A1 )P (A3 |A1 ∩ A2 ). P (A∩B) P (A)

other words P (B|A) = P (B). If A and B are independent, then A and B 0 are also independent.

Bayes' Theorem:

If B1 , B2 , · · · , Bk constitute a partition of a sample space S and P (Bi ) = 6 0 for )·P (A|Br ) i = 1, 2, · · · , k , then for any event A in S such that P (A) 6= 0 P (Br |A) = PkP (BPr(B for r = i )·P (A|Bi ) i=1 1, 2, · · · , k .

Probability distributions: I. Discrete case. Discrete probability distributions (or probability functions): If X is a discrete random

variable, the function given by f (x) = P (X = x) for each x within the range of X is called the probability distribution of X . f (x) is a probability distribution if and only if (1) f (x) ≥ 0, and (2) P x f (x) = 1. If X is a discrete random variable, P the function given by F (x) = P (X ≤ x) = t≤x f (t) for −∞ < x < ∞ where f (t) is the value of the probability distribution of X at t, is called the distribution function of X . If F (x) is a distribution function then (1) F (−∞) = 0, F (∞) = 1, and (2) if a ≤ b, then F (a) ≤ F (b) for any real numbers a and b.

Distribution functions for discrete random variables:

If the range of a random variable X consists of the values x1 < x2 < x3 < · · · < xn , then f (x1 ) = F (x1 ) and f (xi ) = F (xi ) − F (xi−1 ) for i = 2, 3, · · · , n.

II. Continuous case. Continuous probability density functions: A function with values f (x), dened over the set

of all real numbers, is called X if and R b a probability density function of the continuous random variable R∞ only if P (a < X < b) = a f (x)dx. Where f (x) has the properties (1) f (x) ≥ 0, and (2) −∞ f (x)dx = 1. If X is a continuous random variable R xand the value of its probability density at t is f (t), then the function given by F (x) = P (X ≤ x) = −∞ f (t)dt is called the distribution function of X . If F (x) is a distribution function then (1) F (−∞) = 0, F (∞) = 1, and (2) if a ≤ b, then F (a) ≤ F (b) for any real numbers a and b. If f (x) and F (x) are the values of the probability density and the distribution of the continuous random variable X at x, then P (a < X < b) = F (b) − F (a) for any real a and b with a ≤ b, and f (x) = dFdx(x) where the derivative exists.

Distribution functions for continuous random variables:

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t distributions: t probability distributions: If X and Y

are discrete random variables, the function given by f (x, y) = P (X = x, Y = y) for each pair of values (x, y) within the range of X and Y is called the t probability distribution of X and Y . f (x, y) is a t probability distributions if and only if (1) P P f (x, y) ≥ 0 and (2) x y f (x, y) = 1. P IfPX and Y are discrete random variables, the function given by F (x, y) = P (X ≤ x, Y ≤ y) = s≤x t≤y f (s, t) for −∞ < x < ∞ and −∞ < y < ∞ where f (s, t) is the value of the t probability distribution of X and Y at (s, t), is called the t distribution function of X and Y . If F (x, y) is the value of the t distribution function of two discrete random variables X and Y at (x, y), then (1) F (−∞, −∞) = 0, (2) F (∞, ∞) = 1, (3) if a < b and c < d, then F (a, c) ≤ F (b, d). If X and Y are discrete random variables P and f (x, y) is the value of their t probability distribution at (x, y), the function given by g(x) = y f (x, y) for each x within the P range of X is called the marginal distribution of X . Correspondingly, the function given by h(y) = x f (x, y) for each y within the range of Y is called the marginal distribution of Y . If f (x, y) is the value of the t probability distribution of the discrete random variables X and Y at (x, y), and h(y) is the value of the marginal distribution of Y at y , the (x,y) function given by f (x|y) = fh(y) (h(y) 6= 0) for each x within the range of X , is called the conditional distribution of X given Y = y . Correspondingly, if g(x) is the value of the marginal distribution of X (x,y) at x, the function w(y|x) = fg(x) (g(x) 6= 0) for each y within the range of Y , is called the conditional distribution of Y given X = x. Suppose that X and Y are discrete random variables. If the events X = x and Y = y are independent events for all x and y , then we say that X and Y are independent random variables. In such case, P (X = x, Y = y) = P (X = x) · P (Y = y) or equivalently f (x, y) = f (x|y) · h(y) = g(x) · h(y). Conversely, if for all x and y the t probability function f (x, y) can be expressed as the product of a function of x alone and a function of y alone (which are then the marginal probability functions of X and Y ), X and Y are independent. If, however, f (x, y) cannot be so expressed, then X and Y are dependent.

t distribution functions:

Marginal probability functions: Conditional distribution:

Independent random variables:

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Mathematical expectation: I. Discrete case. Let f (x) be the probability distribution of X then the expected value of X is P

E(X) = x xf (x). Let X be a discrete Prandom variable with probability distribution f (x), the expected value of g(X) is given by E[g(X)] = x g(x)f (x). Let f (x) be the probability density function of X then the expected value R∞ of X is E(X) = −∞ xf (x)dx. Let X be a continuous random variable with probability density f (x), R∞ the expected value of g(X) is given by E[g(X)] = −∞ g(x)f (x)dx.

II. Continuous case.