An unknown quantity Y has a Galenshore(a, ) distribution if its density is given by P(y) = 2/(a)* theta^(2a) * y^(2a-1)* e^(-theta*y^2) For y>0,theta>0 and a>0. Assume for now that a is known. For this density.: E[Y]=(a+1/2) / (theta*(a), E[Y^2]=a/theta^2

A) Identify a class of conjugate prior densities for theta. Plot a few members of this class of densities.

B)Let Y1Yn~i.i.d Galenshore(a,theta). FInd the posterior distribution of theta given Y1,...Yn, Using a prior from your conjugate class.

C)Write down p(theta a | Y1...Yn)/p*(theta b | Y1...Yn) and simplify. Identify a sufficient statistic.

D)Determine E[theta | y1...yn]