426-001_hw1 - 2009 Fall Stat 426 : Homework 1 Moulinath...

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2009 Fall Stat 426 : Homework 1 Moulinath Banerjee University of Michigan Announcement: The homework carries 50 points and contributes 5 points to the total grade. Your score on the homework is scaled down to out of 5 and recorded. 1. A geometric random variable W takes values { 1, 2, 3, . . . } and P ( W = j ) = θ (1 - θ ) j - 1 ,where 0 < θ < 1. (a) Prove that for any two positive integers i , j , it is the case that, P ( W > i + j | W > i ) = P ( W > j ) . (b) Indeed, the converse is also true. We show that if W is a discrete random variable taking values { 1 , 2 , 3 , . . . } with probabilities { p 1 ,p 2 ,p 3 , . . . } and satisfies the memoryless property, then W must follow a geometric distribution. Follow these steps to establish the fact that W is geometric. Using the fact that W has the memoryless property, show that P ( W > m ) = ( P ( W > 1)) m , for any m 2. As a first step towards proving this show that P ( W > 2) = ( P ( W > 1)) 2 Define θ = P ( W = 1) and 1 - θ = P
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426-001_hw1 - 2009 Fall Stat 426 : Homework 1 Moulinath...

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