# 09_27_1 - STAT 400 Examples for(1 Fall 2011 Gamma...

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Unformatted text preview: STAT 400 Examples for 09/27/2011 (1) Fall 2011 Gamma Distribution : ( ) ( ) x e x x f Î» 1 Î± Î± Î± Î»-- Î“ = , 0 â‰¤ x < âˆž OR ( ) ( ) Î¸ 1 Î± 1 Î± Î¸ Î± x e x x f-- Î“ = , 0 â‰¤ x < âˆž If T Î± has a Gamma ( Î± , Î¸ = 1 / Î» ) distribution, where Î± is an integer, then F T ( t ) = P ( T Î± â‰¤ t ) = P ( X t â‰¥ Î± ) , P ( T Î± > t ) = P ( X t â‰¤ Î± â€“ 1 ) , where X t has a Poisson ( Î» t = t / Î¸ ) distribution. 1. Let Y be a random variable with a Gamma distribution with Î± = 5 and Î¸ = 3. Find the probability P ( Y > 18 ) â€¦ a) â€¦ by integrating the p.d.f. of the Gamma distribution; P ( Y > 18 ) = ( ) âˆ« âˆž Î“-- â‹… â‹… â‹… 18 3 1 5 5 3 5 1 dx e x x = âˆ« âˆž- â‹… â‹… 18 3 4 5,832 1 dx e x x = â€¦ b) â€¦ by using the relationship between Gamma and Poisson distributions; P ( Y > 18 ) = P ( X 18 â‰¤ 4 ) = 0.285 , where X 18 is Poisson ( 18 / Î¸ = 6 ). EXCEL: = POISSON( x , Î» , ) gives P( X = x ) = POISSON( x , Î» , 1 ) gives P( X â‰¤ x ) A B A B 1 =POISSON(4,18/3,1) â‡’ 1 0.285057 2 2 2. During a radio trivia contest, the radio station receives phone calls according to Poisson process with the average rate of five calls per minute. Find the probability that the ninth phone call would arrive during the third minute. X t = number of phone calls in t minutes. Poisson ( Î» t ) T k = time of the k th phone call. Gamma, Î± = k . five calls per minute â‡’ Î» = 5. a) Find the probability that we would have to wait more than two minutes for the ninth phone call. P ( T 9 > 2 ) = P ( X 2 â‰¤ 8 ) = P ( Poisson ( 10 ) â‰¤ 8 ) = 0.333 . OR P ( T 9 > 2 ) = ( ) âˆ« âˆž Î“-- 2 5 1 9 9 9 5 dt t t e = âˆ« âˆž- 2 5 8 9 8 5 ! dt t t e = â€¦ b) Find the probability that the ninth phone call would arrive during the third minute. P ( 2 < T 9 < 3 ) = P ( T 9 > 2 ) â€“ P ( T 9 > 3 ) = P ( X 2 â‰¤ 8 ) â€“ P ( X 3 â‰¤ 8 ) = P ( Poisson ( 10 ) â‰¤ 8 ) â€“ P ( Poisson ( 15 ) â‰¤ 8 ) = 0.333 â€“ 0.037 = 0.296 . OR P ( 2 < T 9 < 3 ) = ( ) âˆ«-- Î“ 3 2 5 1 9 9 9 5 dt t t e = âˆ«- 3 2 5 8 9 8 5 !...
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09_27_1 - STAT 400 Examples for(1 Fall 2011 Gamma...

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