If $X_i$ i=1,...,n are Gamma($\alpha,\beta$) random variables with known $\alpha$, then
a) find the posterior distribution of $\lambda=1/\beta$ given $X=(X_1,...,X_n)$ if the improper prior distribution of $\lambda$ is $c/\lambda$, where c is a constant.
b) Find the Bayes estimator of $\beta$.
c) Compare the Bayes estimatorof $\beta$ in b) with the UMVUE (uniformly minimum variance unbiased estimator) of $\beta$.

To be certain, we use the following gamma P.D.F.:$$\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$

Question

If $X_i$ i=1,...,n are Gamma($\alpha,\beta$) random variables with known $\alpha$, then 
a) find the posterior distribution of $\lambda=1/\beta$ given $X=(X_1,...,X_n)$ if the improper prior distribution of $\lambda$ is  $c/\lambda$, where c is a constant.
b) Find the Bayes estimator of $\beta$. 
c) Compare the Bayes estimatorof $\beta$ in b) with the UMVUE (uniformly minimum variance unbiased estimator) of $\beta$.

To be certain, we use the following gamma P.D.F.:$$\frac{x^{\alpha-1}e^{-x/\beta}}{\Gamma(\alpha)\beta^\alpha}$$

anonymous · Answer

Please help :(