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Week09Notes - Stat 311 Introduction to Mathematical...

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Unformatted text preview: Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA October 27-29, 2009 Example • Let X 1 , X 2 , ..., X n be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k . Let Y denote the minimum of the X i ’s. – Find the distribution of Y . Geometric Distribution • The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ... } , • the probability distribution of the number Y = X- 1 of failures before the first success, supported on the set { 0, 1, 2, 3, ... } . • Which of these one calls “the” geometric distribution is a matter of convention and convenience. We use the first one. Example • T be the number of trials up to and including the first success. Then P ( T = 1) = p , P ( T = 2) = qp , P ( T = 3) = q 2 p , and in general, P ( T = n ) = q n- 1 p . “Memoryless” property • It is often assumed that the length of time required to service a customer also has a geometric distribution but with a different value for p . P ( T > r + s | T > r ) = P ( T > r + s ) P ( T > r ) = q r + s q r = q s . • Thus, the probability that the customer’s service takes s more time units is independent of the length of time r that the customer has already been served. Because of this interpretation, this property is called the “memoryless” property, and is also obeyed by the exponential distribution. Example • X is geometric( p ), compute P (5 ≤ X ≤ 9). Negative Binomial Distribution • Suppose we are given a coin which has probability p of coming up heads when it is tossed. • We fix a positive integer k , and toss the coin until the k th head appears. • We let X represent the number of tosses. • When k = 1, X is geometrically distributed. • For a general k , we say that X has a negative binomial distribution. Formula • If X = x , then it must be true that there were exactly k- 1 heads thrown in the first x- 1 tosses, and a head must have been thrown on the x th toss. There are x- 1 k- 1 ¶ sequences of length x with these properties, and each of them is assigned the same probability, namely p k- 1 q x- k . • Therefore, if we define u ( x,k,p ) = P ( X = x ) , then u ( x,k,p ) = x- 1 k- 1 ¶ p k q x- k ....
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Week09Notes - Stat 311 Introduction to Mathematical...

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