MHB How to Find the Conditional Density for an Improved Estimator?

[Fermat1]

Consider a family of densitites $f(x,\theta)=\frac{exp(-{\sqrt{x}})}{{\theta}}$. Let $X_{1}$ be a single observation from this family. I have shown that ${\sqrt{X_{1}}}/2$ is an unbiased estimator. Now consider $n$ observations $X_{1},..X_{n}$. I have shown that $T(X)={\sqrt{X_{1}}}+..+{\sqrt{X_{n}}}$ is a sufficient statistic for $\theta$. Now use the Rao Blackwell theorem to find an improved estimator.

The improved estimator is $E({\sqrt{x}}/2|T)$. For this I need the conditional density. How do I find this?

Thanks

 

[stainburg]

Are you sure $$f(x,\theta)$$ is a family of densities?

 

[Fermat1]

Are you sure $$f(x,\theta)$$ is a family of densities?

I got the density completely wrong. Anyway I have just found a way to do the question.