Stat 135, Fall 2006 A. Adhikari HOMEWORK 3 (due Friday 9/22) 1. Read Section 8.2 and Example A of Section 8.4. There’s some redundancy there, but it doesn’t hurt to re-read stuﬀ about the Poisson. Then do Problem 8.10. 2. 8.2. You’ve already done the necessary reading. 3. 8.4a,b. 4. This problem asks you to establish three facts stated in lecture. Recall that the gamma density with parameters α > 0 and λ > 0 is deﬁned by λ α Γ( α ) t α-1 e-λt , t >0 The denominator in the constant of integration is Γ( α ) = Z ∞0 t α-1 e-t dt a) Use integration by parts to show that Γ( α + 1) = α Γ( α ) for all α > 0. b) Let Z have the standard normal density. Show that Z 2 has the gamma density with param-eters α = 1 / 2 and λ = 1 / 2 (a.k.a. the chi-squared density with 1 degree of freedom). How can you use your calculation to show that Γ(1 / 2) = √ π ? In Problems 6 to 10, please use R wherever possible. These problems are intended to give you a warmup with the package. 6.
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