discretedistributions

# discretedistributions - Discrete Distributions Enrique...

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Discrete Distributions Enrique Campos-N´ a˜nez The George Washington University ApSc 116 Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 1 / 32

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Table of Contents 1 Bernoulli 2 Binomial 3 Hypergeometric Distribution 4 Poisson Distribution 5 Negative Binomial 6 The Geometric Distribution Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 2 / 32
Bernoulli The Bernoulli Distribution The random variable X has a Bernoulli distribution with parameter p , with 0 p 1 if X has possible values 1 and 0, Pr ( X = 1) = p , and Pr ( X = 0) = 1 - p q . We can write the p.f. of X as f ( x ) = f ( x | p ) = ± p x q (1 - x ) for x = 0 , 1 , 0 otherwise. Result E ( X k ) = p, and E ( X ) = p and Var ( X ) = pq. Moment generation function ψ ( t ) = ( pe t + q ) . Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 3 / 32

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Binomial The Binomial Distribution Deﬁnition: X has a binomial distribution with parameters n and p , and we write X B ( n , p ), if f ( x | n , p ) = ± ( n x ) p x q n - x for x = 0 , 1 ,..., n , 0 otherwise. If the random variables in a sequence X 1 , X 2 ,... are i.i.d. and each has a Bernoulli distribution with parameter p , then we say that X 1 , X 2 , . . . form a sequence of Bernoulli trials. The i th trial yields r.v. X i and thus a value of either 1 or 0. If the random variables X 1 . . . X n form n Bernoulli trials with parameter p , and if X n i =1 X i , then X B ( n , p ). Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 4 / 32
Binomial Results for Binomial Using the fact that E ( X 1 + ··· + X n ) = E ( X 1 ) + ··· + E ( X n ), we get that E ( X ) = np , Var ( X ) = npq . The binomial’s m.g.f. takes the form ψ ( t ) = E ( e tX ) = n Y i =1 E ( e tX i ) = ( pe t + q ) n . Result (Sum of Binomials) If X 1 ,. . . ,X k are independent, and each X i B ( n i , p ) , and we set X = k i =1 X i , then X B ( n , p ) , n = n 1 + ··· + n k . Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 5 / 32

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Binomial Problem (Exercise: Page 251, # 6) Three men A, B, and C shoot at a target. Suppose that A shoots three times and the probability that he will hit the target on any given shot is 1/8, B shoots ﬁve times and the probability that he will hit the target on any given shot is 1/4, and C shoots twice and the probability that he will hit the target on any given shot is 1/2. What is the expected number of times that the target will be hit? Campos-N´ a˜nez (GWU) Discrete Distributions ApSc 116 6 / 32
Binomial Problem (Exercise: Page 251, # 6) Three men A, B, and C shoot at a target. Suppose that A shoots three times and the probability that he will hit the target on any given shot is 1/8, B shoots ﬁve times and the probability that he will hit the target on any given shot is 1/4, and C shoots twice and the probability that he will hit the target on any given shot is 1/2. What is the expected number of times that the target will be hit?

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## This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.

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discretedistributions - Discrete Distributions Enrique...

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