Let $X_1,X_2,\ldots,X_n$ be a random sample with $n\geq 2$ from an exponential distribution. $X_{(1)}=\min(X_1,X_2,\ldots,X_n)$. Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$.
I was able to find the Cdf of $X_{(1)}$ which is
\begin{align}
F_{X_{(1)}}(x)&=
\begin{cases}
0, & x<0
\\
1-e^{-n \lambda x}, & x\geq 0
\end{cases}\label{6}
\end{align}
But I am unable to understand how to proceed from here. Please help.
Best Answer
Assuming $X_1,X_2,\ldots,X_n$ are i.i.d Exponential with mean $1/\lambda$, the conditional expectation can be found indirectly using the theory of minimum variance unbiased estimation.
As $T=\sum\limits_{i=1}^n X_i$ is a complete sufficient statistic for this family of distributions, $E\left[X_{(1)}\mid T\right]$ is simply the uniformly minimum variance unbiased estimator (UMVUE) of $E\left[X_{(1)}\right]$ by Lehmann-Scheffé theorem.
But note that $X_{(1)}$ is Exponential with mean $\frac{1}{n\lambda}$ and
$$E\left[\frac1{n^2}\sum_{i=1}^n X_i\right]=\frac1{n^2}\sum_{i=1}^n \frac1{\lambda}=\frac1{n\lambda}$$
By Lehmann-Scheffé, this again suggests that $\frac1{n^2}\sum\limits_{i=1}^n X_i$ is the UMVUE of $E\left[X_{(1)}\right]=\frac1{n\lambda}$.
As UMVUE is unique whenever it exists, it must be that
$$E\left[X_{(1)}\mid T\right]=\frac{T}{n^2} \quad,\text{ a.e.}$$