Let $X_1,..,X_n$ random sample of $X\sim\text{Exp}(\lambda)$ with $f(x;\lambda)=\frac{1}{\lambda}e^{-\frac{1}{\lambda}x}I_{[0,\infty]}(x)$
i) Find a unbiased estimator of $\lambda$ based only in $X_{(1)}=\min(X_i)$
ii) Apply the Rao-Blackwell theorem for find a estimator better that you find in i)
For i) I find that $\hat{\lambda}=nX_{(1)}$ is unbiased estimator for $\lambda$. Now for the part ii) that is the problem.
I know that $f(x;\lambda)\in$ the exponential family and $T=\sum X_i$ is a complete and sufficient statistic. Then since $\phi_T=E[nX_{(1)}|\sum X_i]$ produces a unbiased estimator, then it needs to be UMVUE.
So I calculated the Cramer-Rao Lower Bound and find $\text{var}_\lambda\geq \frac{\lambda^2}{n}$, finally I just take $\phi_T=\frac{\sum X_i}{n}$
Now my doubts are: i)My reasoning is correct?
ii)I always need to calculate conditional distribution?
iii)Is there any simple way to find this conditional?
iv)How could I find the conditional in this case?
I am unable to find that conditional