If $Z = \min(X,Y)$ and $W = \max(X,Y)$, then for $w > z$, $\begin{align*} F_{Z,W}(z,w) &= P\{Z \leq z, W \leq w\}\\ &= P\left[\{X \leq z, Y \leq w\} \cup \{X \leq w, Y \leq z\}\right]\\ &= P\{X \leq z, Y \leq w\} + P\{X \leq w, Y \leq z\} - P\{X \leq z, Y \leq z\}\\ &= F_{X,Y}(z, w) + F_{X, Y}(w,z) - F_{X,Y}(z,z) \end{align*} $ while for $w < z$, $\begin{align*} F_{Z,W}(z,w) &= P\{Z \leq z, W \leq w\} = P\{Z \leq w, W \leq w\}\\ &= P\{X \leq w, Y \leq w\}\\ &= F_{X,Y}(w,w). \end{align*} $ Consequently, if $X$ and $Y$ are jointly continuous random variables, then $f_{Z,W}(z,w) = \frac{\partial^2}{\partial z \partial w}F_{Z,W}(z,w) = \begin{cases} f_{X,Y}(z,w) + f_{X,Y}(w,z), & \text{if}~w > z,\\ \\ 0, & \text{if}~w < z. \end{cases} $ The conditional density of $W$ given $Z = z$ is $ f_{W \mid Z}(w \mid z) = \frac{f_{Z,W}(z,w)}{f_Z(z)} = \begin{cases} \frac{f_{X,Y}(z,w) + f_{X,Y}(w,z)}{\int_z^{\infty} f_{X,Y}(z,w) + f_{X,Y}(w,z)\ \mathrm dw}, & w > z,\\ 0, & w < z, \end{cases} $ and so with $f_{X,Y}(x,y) = e^{-x-y}$ for $x, y \geq 0$ $ \begin{align*}E[W \mid Z = z] &= \frac{\int_z^\infty w\cdot f_{X,Y}(z,w) + w\cdot f_{X,Y}(w,z)\ \mathrm dw}{ \int_z^\infty f_{X,Y}(z,w) + f_{X,Y}(w,z)\ \mathrm dw}\\ &= \frac{\int_z^\infty w\cdot e^{-w-z} + w\cdot e^{-w-z}\ \mathrm dw}{ \int_z^\infty e^{-w-z} + e^{-w-z}\ \mathrm dw}\\ &= \frac{2e^{-2z}\int_z^\infty w\cdot e^{-w}\ \mathrm dw}{ 2e^{-2z}} = \frac{2e^{-z}[\left . (-we^{-w})\right\vert_z^{\infty} + \int_z^{\infty}e^{-w}\ \mathrm dw]}{2e^{-2z}}\\ &= 1 + z. \end{align*} $