# h9 - Stat 5101(Geyer Fall 2011 Homework Assignment 9 Due...

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Stat 5101 (Geyer) Fall 2011 Homework Assignment 9 Due Wednesday, November 23, 2011 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 9-1. Suppose E ( Y | X ) = X var( Y | X ) = 3 X 2 and suppose the marginal distribution of X is N ( μ, σ 2 ). (a) Find E ( Y ). (b) Find var( Y ). 9-2. Suppose X 1 , . . . , X N are IID having mean μ and variance σ 2 where N is a Poi( λ ) random variable independent of all of the X i . Let Y = N X i =1 X i , with the convention that N = 0 implies Y = 0. (a) Find E ( Y ). (b) Find var( Y ). 9-3. Suppose that the conditional distribution of Y given X is Poi( X ), and suppose that the marginal distribution of X is Gam( α, λ ). Show that the marginal distribution of Y is a negative binomial distribution in the extended sense discussed in the brand name distributions handout in which the shape parameter need not be an integer. Identify the parameters of this

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