Let $F$ be exponential, find $\Bbb E [ X_j | X_i ]$, where $X_i$ and $X_j$ are the order statistic $i<j$.
I understand that to find $f(X_j|X_i)$ I can use the conditional definition of $\frac{f(X_i,X_j)}{f(X_i)}$. I end up with:
$$f(X_j|X_i)=\frac{(n-i)!f(X_i)[FX_j-FX_i]^{j-i-1} [1-FX_j]^{n-j}}{(j-i-1)!(n-j)![1-FX_j]^{n-i}}$$
Now, I need to plug in $F$~exponential and get the expected value but I'm not quite sure how to do this. Any guidance will be appreciated.