Now, suppose that we instead have the proper prior ĢĢ(Ģ²Ģ) ~ Exp(a) (a > 0). Again, just as in part (b): conditional on Ģ²Ģ, we have observations Xā, Xā, ..., Xā i.i.d N(Ģ²Ģ, 1). You may assume that a < āXį. Compute the posterior distribution ĢĢ(Ģ²Ģ | Xā, Xā, ..., Xā), then provide the following statistics on the posterior distribution. Write Phi for the CDF function Ī¦() and PhiInv for its inverse.
Use SumXi for āįāįį Xį.
median:
1/n*SumXi-a*n
mode:
n*SumXi-a)n