Let $X_{1},...,X_{n}$ be a random sample of size n from the continuous distribution with pdf:
$f_{X}(x|\alpha,\beta) = \frac{2*\beta^{\alpha}}{\Gamma(\alpha)}*(\frac{1}{x})^{2*\alpha+1}*\exp({\frac{-\beta}{x^{2}}})*I_{(0,\infty)}(x)$
where $\alpha$ > 0 is fixed and $\beta$ > 0. Assume the prior distribution on $\beta$ has the pdf:
$\pi(\beta|\lambda) = \lambda*e^{-\lambda*\beta}$
where $\lambda$ > 0 is fixed. Find the Bayes estimator of $\beta$.
So to find the Bayes estimator of $\beta$ I know I need to find the posterior distribution, which I did, and I got
$\pi(\Theta|X)$ = $e^{\frac{\lambda*\beta*x^{2}-\beta}{x^{2}}*2*\beta^{\alpha}}$
which I am questioning if it is right, especially since I'm not sure my marginal distribution of x is right, either, but I thought that since both $\alpha$ and $\lambda$ were fixed, then I could treat them like constants, but I wasn't sure what I need to replace in the joint to make sure that it integrated to 1 so then it just 'disappeared' because it integrated to 1. Also, to find the Bayes estimator I also know that I need to take the conditional expected value of the posterior distribution, but when I'm not sure if my posterior is correct, I didn't want to start that. Any help would be greatly appreciated!