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410Hw05ans - STAT 410 Summer 2009 Homework#5(due Friday...

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STAT 410 Summer 2009 Homework #5 (due Friday, July 10, by 4:00 p.m.) 1. Let X and Y be independent random variables, each geometrically distributed with the probability of “success” p , 0 < p < 1. That is, p X ( k ) = p Y ( k ) = ( 29 1 1 - - k p p , k = 1, 2, 3, … , a) Find P ( X > Y ) . [ Hint: You may find it helpful to find P ( X = Y ) first. ] P ( X = Y ) = ( 29 ( 29 = 1 Y X k k p k p = ( 29 ( 29 - - = - - 1 1 1 1 1 k k k p p p p = ( 29 [ ] - = - 1 1 2 2 1 k k p p = ( 29 [ ] - = 0 2 2 1 n n p p = ( 29 2 2 1 1 p p - - = p p - 2 . P ( X > Y ) + P ( X = Y ) + P ( X < Y ) = 1. Since P ( X > Y ) = P ( X < Y ) , P ( X > Y ) = ( 29 ( 29 Y X P 1 2 1 = - = - - 2 1 2 1 p p = p p - - 2 1 . OR P ( X > Y ) = ( 29 ( 29 - - = + = - - 1 1 1 1 1 1 y y x y x p p p p = ( 29 ( 29 - - = + = - - 1 1 1 1 2 1 1 y y x x y p p p = ( 29 ( 29 ( 29 - - - - = - 1 1 2 1 1 1 1 y y y p p p p = ( 29 - = - 1 1 2 1 y y p p = ( 29 ( 29 [ ] - - = 0 2 1 1 n n p p p = ( ( 29 2 1 1 1 p p p - - - = p p - - 2 1 .

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b) Find P ( X + Y = n ) , n = 2, 3, 4, … , and P ( X = k | X + Y = n ) , k = 1, 2, 3, … , n – 1. P ( X + Y = n ) = ( 29 ( 29 - = - = = 1 1 Y P X P n k k n k = ( 29 ( 29 - = - - - - - 1 1 1 1 1 1 n k k n k p p p p = ( 29 - = - - 1 1 2 2 1 n k n p p = ( 29 ( 29 2 2 1 1 - - - n p p n , n = 2, 3, 4, … . Since X and Y both have Geometric ( p ) distribution and are independent, X + Y has Negative Binomial distribution with r = 2. P ( X = k | X + Y = n ) = ( ( 29 Y X P Y X X P n n k = + = + = = ( ( 29 Y X P Y X P n k n k = + - = = = ( 29 ( 29 ( 29 ( 29 2 2 1 1 1 1 1 1 - - - - - - - - n k n k p p n p p p p = 1 1 - n , k = 1, 2, 3, … , n – 1.
2. Suppose the size of largemouth bass in a particular lake is uniformly distributed over the interval 0 to 8 pounds. A fisherman catches (a random sample of) 5 fish. X 1 , X 2 , X 3 , X 4 , X 5 Y k = k th smallest. First find F X ( x ) = P ( X x ) F X ( x ) = 8 8 1 0 x x dy = , 0 < x < 8. a) What is the probability that the smallest fish weighs less than 2 pounds? P ( Y 1 < 2 ) = 1 – P ( Y 1 > 2 ) . P ( Y 1 > 2 ) = P ( X 1 > 2, X 2 > 2, X 3 > 2, X 4 > 2, X 5 > 2 ) = ( 6 / 8 ) 5 0.2373. P ( Y 1 < 2 ) = 1 – P ( Y 1 > 2 ) = 1 – ( 6 / 8 ) 5 0.7627 . OR f min X i ( x ) = ( 29 ( 29 ( 29 1 F 1 x f x n n - - = 5 ( 1 – x / 8 ) 4 ( 1 / 8 ) , 0 < x < 8. P ( Y 1 < 2 ) = ( 29 2 0 X min dx x f i . b) What is the probability that the largest fish weighs over 7 pounds? P ( Y 5 > 7 ) = 1 – P ( Y 5 < 7 ) . P ( Y 5 < 7 ) = P ( X 1 < 7, X 2 < 7, X 3 < 7, X 4 < 7, X 5 < 7 ) = ( 7 / 8 ) 5 0.5129. P ( Y 5 > 7 ) = 1 – P ( Y 5 < 7 ) = 1 – ( 7 / 8 ) 5 0.4871 . OR f max X i ( x ) = ( 29 ( 29 ( 29 1 F x f x n n - = 5 ( x / 8 ) 4 ( 1 / 8 ) = 5 x 4 / 8 5 , 0 < x < 8. P ( Y 5 > 7 ) = ( 29 7 8 5 5 8 7 5 4 8 7 X 8 8 5 max x dx x dx x f i = = = 1 – ( 7 / 8 ) 5 0.4871 .

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c) What is the probability that the largest fish weighs between 6 and 7 pounds?
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410Hw05ans - STAT 410 Summer 2009 Homework#5(due Friday...

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