Let $X_1,X_2,...,X_n$ be a random sample of size $n \geq 2$ from $\exp(1/\theta)$, where $\theta \in (0,\infty)$.
Let $$Y = \min\{X_1,X_2,...,X_n\} \qquad \textrm{and} \qquad T = \sum_{i=1}^{n} X_i .$$ I am supposed to find $E[Y|T]$.
I tried using the Basu's theorem, but then I realized that I could apply it only when $T$ took a particular value. Also, I tried finding the joint pdf of first order stats and $T$, but couldn't do that either.