# stat400lec11 - shots(4 What is the probability that he must...

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Statistics 400 Lecture 11 Section 3.4 The Moment-generating function Moment-generating function M(t)=E[e tX ]= ) ( x f e tx M’(t)= ) ( x f xe tx M’(0)= ) ( x f x =E[X] M”(t)= ) ( 2 x f e x tx M”(0)= ) ( 2 x f x =E[X 2 ] Thus Var[X]=E[X 2 ]-(E[X]) 2 Example : The mean of the binomial distribution f(x)= x n x p p x n - - ) 1 ( M(t)=E(e tX ) Ping Ma Lecture 11 Fall 2005 - 1 -

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Example: According to NBA.com’s statistics, Yao Ming can make a free throw 80% of the time. Assume that free throw can be thought of as independent Bernoulli trials. (1) What is the probability that he must attempt 15 throws to make a total 10 shots? (2) What is the probability that he must attempt 15 throws to make a total 10 shots? Ping Ma Lecture 11 Fall 2005 - 2 -
(3) What is the probability that he must attempt x throws to make a total 10

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Unformatted text preview: shots? (4) What is the probability that he must attempt x throws to make a total r shots? Negative binomial distribution Ping Ma Lecture 11 Fall 2005 - 3 -Binomial distribution f( x )= x n x p p x n-- ) 1 ( When n is very large and p is very small, we have difficulty to calculate the pmf of binomial distribution. If λ = np , we can approximate the binomial distribution ! 1 )! ( ! ! lim ) 1 ( lim x e n n x n x n p p x n x x n x n x n x n--∞ →-∞ → = - -=- Poisson distribution ! ) ( x e x f x-= , x=0,1,2,…. E[X] = Var[X]= Ping Ma Lecture 11 Fall 2005 - 4 -...
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## This note was uploaded on 07/24/2008 for the course STAT 400 taught by Professor Tba during the Fall '05 term at University of Illinois at Urbana–Champaign.

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stat400lec11 - shots(4 What is the probability that he must...

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