Conditional expectation of $\max(X,Y)$ and $\min(X,Y)$ when $X,Y$ are iid and exponentially distributed I am trying to compute the conditional expectation $$E[\max(X,Y) | \min(X,Y)]$$ where $X$ and $Y$ are two iid random variables with $X,Y \sim \exp(1)$. 
I already calculated the densities of $\min(X,Y)$ and $\max(X,Y)$, but I failed in calculating the joint density. Is this the right way? How can I compute the joint density then? Or do I have to take another ansatz? 

You can presumably find $\Pr(X\le a)$ and $\Pr(b \lt X \le a) $ and so  $\Pr(b \lt X \le a) \Pr(b \lt Y \le a)$.
Take the derivatives of this with respect to $a$ then and $b$ (changing the sign as $b$ is a lower limit) and add an indicator such as $I_{b\le a}$, and you have the joint density which we might call $p(a,b)$ with $a$ acting as $\max(X,Y)$ and $b$ acting as $\min(X,Y)$.
You can then work out the conditional density $p(a|b) =\dfrac{p(a,b)}{\int_{a=b}^\infty p(a,b) \; da}$ and the conditional mean $E[a|b] = \int_{a=b}^\infty a\; p(a|b) \; da$, which will be a function of $b$. 
To check, remember that the exponential distribution is memoryless so if $X$ and $Y$ have mean $\mu$ then you will have $E[\max(X,Y) | \min(X,Y)] = \min(X,Y) +\mu$.
