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Unformatted text preview: 36-226 Summer 2010Homework 9Solutions1.p(θ|X1,...,Xn)∝nYi=11θe-xi/θβαΓ(α)1θα+1e-β/θ∝1θnexp(-1θnXi=1xi)1θα+1e-β/θ∝1θn+α+1exp(-1θ(nXi=1xi+β))∝Inv-Gamma(n+α,nXi=1xi+β)Thus, since the prior and posterior both come from the Inv-Gamma family, Inv-Gamma isconjugate for exponential.2.(a) I choseα= 21 andβ= 200. This gives me a mean ofθ= 10 and sd of 2.3. Thus Ifeel like the average waiting time should be somewhere between 7.7 minutes and 12.3minutes. The prior is shown in Figure1. Remember that this is MY prior. Yours willbe different. I will show all the work for the rest of the assignment usingαandβandplugging in my values at the end.51015200.000.050.100.150.20a=21, b=200mean=10, sd=2.294xyFigure 1: Prior distribution forθ(b) As we found in problem 1, the posterior distribution is also Inv-Gamma with parametersα=n+αandβ=∑ni=1xi+β. Therefore MY posterior isp(θ|X1,...,Xn) = Inv-Gamma(α= 10 +α,β= 164.5 +β)= Inv-Gamma(α= 31,β= 364.5)....
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- Summer '09