Exponential arrivals are dependent on a parameter;fx(x)=e(x);>0,x>0 Two interarrival times are recorded and an estimate of is to be made. However, expert judgment was first captured as the prior distribution f()=5/4(2);15. We letW=X1+X2 for independentX1,X2. First report the distribution forfw(w) as the sum of exponentials (a known result, just reference) and then

[a] Establish an expression for the posterior distributiong(W=k)

Then assumingX1=0.25,X2=0.75

[b] Calculate the maximum likelihood estimate for , using a known result

[c] Calculate the Bayes estimate for assuming the loss functionL((hat),)=((hat))2

[d] Calculate the Bayes estimate for assuming the loss functionL((hat),)=(hat)