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Assignment 7

Assignment 7 - Stat 5101(Geyer Fall 2009 Homework...

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Stat 5101 (Geyer) Fall 2009 Homework Assignment 7 Due Friday, November 6, 2009 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. 7-1. If X has the Gam( α, λ ) distribution, we calculated in class that E ( X β ) = Γ( α + β ) Γ( α ) λ β . (a) Find E ( X 2 ) (b) Find var( X ). None of your answers should contain gamma functions (use the gamma function recursion formula to simplify). 7-2. If X has the Beta( α 1 , α 2 ) distribution, show that E { X β 1 (1 - X ) β 2 } = Γ( α 1 + α 2 )Γ( α 1 + β 1 )Γ( α 2 + β 2 ) Γ( α 1 )Γ( α 2 )Γ( α 1 + α 2 + β 1 + β 2 ) Hint: use the fact that the PDF of the beta distribution integrates to one, just like we did for the gamma distribution. You may ignore the issue of when the integral exists (it exists when β 1 > - α 1 and β 2 > - α 2 , but we don’t know how to prove that yet). 7-3. Suppose X has the Beta( α 1 , α 2 ) distribution.

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Assignment 7 - Stat 5101(Geyer Fall 2009 Homework...

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