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# stat400lec15 - y t âˆ âˆž-= Î“ 1 = Î“ n Gamma distribution...

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Statistics 400 Lecture 15 (Sep 28) Waiting Time Review: Example : Customers arrive at Staples according a Poisson distribution with mean rate 1/3 per minute. On Thanksgiving morning, Staples opens 5:00am in the Morning. The first 10 customers will get a free flash drive. Let W denote the waiting time until the 10 th customer comes in. What is the cumulative distribution function of W. F(w)=P(W<w) =1-P(W>w) =1-P(fewer than 10 customers come in between 0 and w minutes) =1- = - 9 0 3 1 ! ) 3 1 ( k w K k e w f(w)=F’(w)= w e w 3 1 9 10 91 3 1 - Example : Customers arrive at Staples according a Poisson distribution with mean rate λ per minute. On Thanksgiving morning, Staples opens 5:00am in the Morning. The first 10 customers will get a free flash drive. Let W denote the waiting time until the α th customer comes in. What is the probability density function of W. f(w)=F’(w)= w e w λ α α α λ - - - 1 )! 1 ( Ping Ma Lecture 15 Fall 2005 - 1 -

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Gamma function
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Unformatted text preview: y t âˆ« âˆž--= Î“ 1 ) ( = Î“ ) ( n Gamma distribution Î¸ Î± / 1 ) ( 1 ) ( x e x x f--Î“ = âˆž < â‰¤ x E[X]= Î±Î¸ Var[X]= 2 Example : Customers arrive at Staples according a Poisson distribution with mean rate 1/2 per minute. Staples opens on 9:00am in the morning. W]hat is the probability that the second customer arrives after 9:05am? Ping Ma Lecture 15 Fall 2005- 2 -Special case of Gamma distribution 2 , 2 r = = Î± Î¸ where r is a positive integer. / 1 ) ( 1 ) ( x e x x f--Î“ = 2 / 1 2 / 2 / 2 ) 2 / ( 1 x r r e x r--Î“ = Chi-square distribution with r degrees of freedom, ) ( 2 r Ï‡ E[X]= Î±Î¸ = r r = 2 2 Var[X]= 2 = r r 2 2 2 2 = 100(1-) th percentile ) ( 2 r 100 th percentile ) ( 2 1 r-Ping Ma Lecture 15 Fall 2005- 3 -...
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