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# 410Hw02ans - STAT 410 Fall 2009 Homework#2(due Friday...

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STAT 410 Fall 2009 Homework #2 (due Friday, September 11, by 3:00 p.m.) 1. The p.d.f. of X is f X ( x ) = θ x θ – 1 , 0 < x < 1, 0 < θ < . Let Y = – 2 θ ln X. How is Y distributed? a) Determine the probability distribution of Y by finding the c.d.f. of Y F Y ( y ) = P ( Y y ) = P ( 2 θ ln X y ) . “Hint”: Find F X ( x ) first. F X ( x ) = x θ , 0 < x < 1. 0 < x < 1 y = – 2 θ ln x y > 0 F Y ( y ) = P ( Y y ) = P ( 2 θ ln X y ) = P ( X e y / 2 θ ) = 1 – e y / 2 , y > 0. f Y ( y ) = F Y ' ( y ) = 2 1 e y / 2 , y > 0. Y has Exponential distribution with mean 2 . b) Determine the probability distribution of Y by finding the m.g.f. of Y M Y ( t ) = E ( e Y t ) = E ( e 2 θ ln X t ) . M Y ( t ) = E ( e Y t ) = E ( e 2 θ ln X t ) = E ( X 2 θ t ) = ( - - 1 0 1 θ θ 2 θ dx x x t = - - 1 0 1 θ 2 θ θ dx x t = t θ θ θ 2 - = t 2 1 1 - , t < 2 1 . Y has Exponential distribution with mean 2 .

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c) Determine the probability distribution of Y by finding the p.d.f. of Y, f Y ( y ) , using Theorem 1.7.1. y = g ( x ) = 2 θ ln x x = g 1 ( y ) = = e y / 2 θ d x / d y = θ 2 1 - e y / 2 θ f Y ( y ) = f X ( g 1 ( y ) ) y x d d = 2 1 θ θ θ 2 1 2 θ θ y y e e - - - - × = 2 1 e y / 2 , y > 0. Y has Exponential distribution with mean 2 .
2. A fair 6-sided die is rolled repeatedly. Find the probability that … a) the first “6” occurs on an even-numbered attempt; P(even) = P(2) + P(4) + P(6) + … = + + + 5 3 1 6 5 6 1 6 5 6 1 6 5 6 1 = 11 5 = - = = = = + 36 25 1 1 36 5 36 25 36 5 6 5 6 1 0 0 1 2 n n k k . OR P(even) = + + + 5 3 1 6 5 6 1 6 5 6 1 6 5 6 1 P(odd) = + + + 4 2 0 6 5 6 1 6 5 6 1 6 5 6 1 P(even) = 6 5 P(odd). P(odd) = 5 6 P(even). 1 = P(odd) + P(even) = 5 11 P(even). P(even) = 11 5 . b) “6” occurs before an odd number is rolled. P ( “6” before odd ) = = + + + + + = 0 2 6 1 6 2 ... 6 1 6 2 ... 6 1 6 2 6 1 6 2 6 1 k k k = 4 1 6 2 1 1 6 1 = - = 0.25 . 2 / 6 = P( not “6” and not odd ) OR Four “relevant” outcomes: one “good” outcome “6” three “bad” outcomes “1”, “3”, “5” P ( “6” before odd ) = 1 / 4 = 0.25 .

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410Hw02ans - STAT 410 Fall 2009 Homework#2(due Friday...

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