hw9 - 36-226 Summer 2010Homework 9Due Aug 4START EARLY SO...

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Unformatted text preview: 36-226 Summer 2010Homework 9Due Aug. 4START EARLY SO THAT YOU CAN STUDY FOR THE FINAL1. LetX1,...,Xn∼exp(θ). Show that the Inv-Gamma(α,β) is conjugate for exponential andfind the form of the posterior. Use the following forms for the Exponential and Inv-Gammadistributions:p(X|θ) =1θe-x/θp(θ|α,β) =βαΓ(α)1θα+1e-β/θ2. I want to know the typical amount of time that I will wait for a 500 bus at 5thand Moorewood.According to the bus schedule, busses should arrive every 15 minutes or so. Waiting timesare typically modelled as exponential distributions. I therefore decide to check how long Iwait for a few days and use a Inv-Gamma prior to make inferences about the true waitingtime.(a) Choosehyperparameters(the fixed, knownαandβvalues in the Gamma distribution)by using the followingRfunction.fcn = function(a,b){x = seq(0,20,len=200)y = (b^a)/(gamma(a)) * x^(-a-1) * exp(-b/x)plot(x,y,main=paste(’a=’,a,’, b=’,b,’\n mean=’,round(b/(a-1),3),’, sd=’,round(b/((a-1)*sqrt(a-2)),3),sep=’’),ty=’l’)}This function will plot yourpriorbeliefs aboutθfor different values of the hyperparam-...
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hw9 - 36-226 Summer 2010Homework 9Due Aug 4START EARLY SO...

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