# 410Hw10ans - STAT 410 Homework#10(due Friday July 31 by...

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STAT 410 Homework #10 Spring 2009 (due Friday, July 31, by 4:00 p.m.) 1. Suppose X 1 , X 2 , … , X n are independent random variables, and X i has Geometric distribution with probability of “success” p i , i = 1, 2, … , n . Let Y = min X i . What is the probability distribution of Y? Hint: Consider P ( X > x ) for a Geometric ( p ) random variable. Let X be a random variable with a Geometric distribution with probability of “success” p . Then P ( X > y ) = ( 29 - + = - 1 1 1 y k k p p = ( 1 – p ) y , y = 0, 1, 2, 3, … . Let y be a positive integer. P ( Y > y ) = P ( X 1 > y ) P ( X 2 > y ) P ( X n > y ) = ( 1 – p 1 ) y ( 1 – p 2 ) y ( 1 – p n ) y = ( 29 y n i i p 1 1 - = , y = 0, 1, 2, 3, … . p Y ( y ) = P ( Y = y ) = P ( Y > y – 1 ) P ( Y > y ) = ( 29 ( 29 1 1 1 1 1 1 - = = - - - y n i i n i i p p , y = 1, 2, 3, … . Y has a Geometric distribution with probability of “success” p = ( 29 = - - n i i p 1 1 1 .

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2. Every year on July 24, Anytown Tigers and Someville Lions play a soccer game. It is always a high-scoring game, the number of goals scored follows a Poisson process with the average rate of one goal per 5 minutes. What is the probability that the fourth goal is scored during the last 10 minutes of the first half? (A soccer game consists of two halves, 45 minutes each.) a) Find the probability that the fourth goal is scored during the last 10 minutes of the first half by integrating the p.d.f. of Gamma distribution. Include the anti-derivative. Gamma distribution, α = 4, 5 minutes λ = 1. ( 29 - - Γ 9 7 1 4 4 1 dx x e x = - 9 7 3 ! 3 1 dx x e x = - 9 7 3 6 1 dx x e x = 7 9 2 1 6 1 2 3 - - - - - - - - x x x x e e e e x x x = ( – 0.021226 ) – ( – 0.081765 ) = 0.060539 . OR Gamma distribution, α = 4, 1 minute λ = 0.2. ( 29 - - Γ 45 35 2 . 0 1 4 4 4 2 . 0 dx x e x = - 45 35 2 . 0 3 4 ! 3 2 . 0 dx x e x = - 45 35 2 . 0 3 3750 1 dx x e x = 35 45 5 1 50 1 750 1 2 . 0 2 . 0 2 . 0 2 2 . 0 3 - - - - - - - - x x x x e e e e x x x = ( – 0.021226 ) – ( – 0.081765 ) = 0.060539 .
b) Find the probability that the fourth goal is scored during the last 10 minutes of the first half by considering the possible number of goals scored in the first 35 and the next 10 minutes. P ( 4th goal scored during the last 10 minutes of the first half ) = P ( 3 goals scored in 35 minutes ) P ( at least 1 goal scored in the next 10 minutes ) + P ( 2 goals scored in 35 minutes ) P ( at least 2 goals scored in the next 10 minutes ) + P ( 1 goal scored in 35 minutes ) P ( at least 3 goals scored in the next 10 minutes ) + P ( 0 goals scored in 35 minutes ) P ( at least 4 goals scored in the next 10 minutes ) Number of goals scored in 35 minutes ~ Poisson ( 7 ) Number of goals scored in the next 10 minutes ~ Poisson ( 2 ) = -

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410Hw10ans - STAT 410 Homework#10(due Friday July 31 by...

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