Homework_6_Solution

Homework_6_Solution - Stat 414 Homework 6 Homework 6...

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Stat 414 Homework 6 Fall 2008 Homework 6 Solution 1. Suppose that Y has a Geometric(p) distribution. (10 points) In class, we showed that E(Y)=1/p. Recall that E(Y(Y-1)) is the first factorial moment and that in class we showed that E(Y(Y-1))= = 1 1 ) 1 ( y y pq y y = = 2 2 ) 1 ( y y q y y pq = = 2 2 2 y y q dq d pq a) Complete this computation to show that E(Y(Y-1))=2q/p 2 2 3 3 2 2 2 2 2 0 2 2 2 2 2 2 2 2 ) 1 ( 2 ) 1 ( 2 1 1)) - E(Y(Y p q p pq q pq q q q dq d pq q q dq d pq q q dq d pq q dq d pq y y y y = = = = = = = = = b) Use the result in part a) to show that Var(Y)=q/p 2 . E[Y(Y-1)] = E(Y 2 ) – E(Y) b E(Y 2 ) = E[Y(Y-1)] + E(Y) = 2q/p 2 + 1/p Var(Y) = E(Y 2 ) – E 2 [Y] = 2q/p 2 + 1/p – 1/p 2 = (2q + p - 1)/p 2 = (q + q + p – 1)/p 2 = = (q + 1 – 1)/p 2 = q/p 2 . c) The moment generating function is m(t) = E(e Yt ) = = 1 y y yt q e q p . For what values of t is the moment generating function defined? (i.e. when does the sum converge?) m(t) = E(e Yt ) = ( ) y qe q p q e q p y t y y yt = = = 1 1 The sum converges if |qe t | < 1 or t < ln(1/q) or t < -lnq d) In class we showed that m(t)= t t qe qe q p 1 . Use this fact to compute E(Y) and Var(Y) by differentiating.
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Stat 414 Homework 6 Fall 2008 p p q q q m qe qe t t / 1 . q p ) 1 ( . q p ) 0 ( ' ) 1 ( . q p (t) m' 2 2 2 = = = = = μ 2 2 2 2 2 3 2 2 3 / 1 1 1 1 1 ) 1 ( . q p ) 0 ( ' ' ) 1 ( ) 1 ( . q p (t) ' m' p q p q p p q p p q q m qe qe qe t t t = + = = + = + == = + = σ e) Show that the cumulative distribution function Y is + < < = 1 1 1 0 ) ( y t y q t t F y where y =1,2,. .. When t < 0, F(t) = P(Y ≤ t) = 0 since Y must have a discrete value.
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Homework_6_Solution - Stat 414 Homework 6 Homework 6...

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