Hw04ans - STAT 410 Spring 2008 Homework #4 1. Let X and Y...

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Unformatted text preview: STAT 410 Spring 2008 Homework #4 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 ) = ( ) 1 1-- ⋅ k p p , k = 1, 2, 3, … , a) Find P ( X > Y ). [ Hint: First, find P ( X = Y ). ] P ( X = Y ) = ( ) ( ) & ∞ = ⋅ 1 Y X k k p k p = ( ) ( ) & ∞-- =-- ⋅ ⋅ ⋅ 1 1 1 1 1 k k k p p p p = ( ) [ ] & ∞- =- ⋅ 1 1 2 2 1 k k p p = ( ) [ ] & ∞- = ⋅ 2 2 1 n n p p = ( ) 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 ) = ( ) ( ) Y X P 1 2 1 =- ⋅ = ¡ ¡ ¢ £ ¤ ¤ ¥ ¦-- ⋅ 2 1 2 1 p p = p p-- 2 1 . OR P ( X > Y ) = ( ) ( ) & & ∞ ∞-- = + =-- ⋅ ⋅ ⋅ 1 1 1 1 1 1 y y x y x p p p p = ( ) ( ) & & ∞ ∞-- = + =-- ⋅ ⋅ 1 1 1 1 2 1 1 y y x x y p p p = ( ) ( ) ( ) & ∞---- =- ⋅ ⋅ 1 1 2 1 1 1 1 y y y p p p p = ( ) & ∞- =- ⋅ 1 1 2 1 y y p p = ( ) ( ) [ ] & ∞-- = ⋅ ⋅ 2 1 1 n n p p p = ( ) ( ) 2 1 1 1 p p p--- ⋅ = p p-- 2 1 . 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 ) = ( ) ( ) &- =- = = ⋅ 1 1 Y P X P n k k n k = ( ) ( ) &- =----- ⋅ ⋅ ⋅ 1 1 1 1 1 1 n k k n k p p p p = ( ) &- =-- ⋅ 1 1 2 2 1 n k n p p = ( ) ( ) 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....
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This note was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw04ans - STAT 410 Spring 2008 Homework #4 1. Let X and Y...

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