Joint distribution of $\max(X, Z)$ and $\max(Y, Z)$ where $X$, $Y$, $Z$ are independent exponential variables with mean $1$ Assume there are three independent exponential random variables $X$, $Y$, and $Z$ with mean $1$. I am trying to look at the joint distribution of $[U = \max(X, Z), V = \max(Y, Z)]$.
I want to find $F_{\, U,V}(u, v)$ which is the cumulative distribution function of joint distribution $(U,V)$.
What I am doing is diving the possible relations of $X$, $Y$, $Z$  like $X < Y < Z$, $X < Z < Y$, etc. But I’m getting stuck at cases like $u < v$ and $Z < X < Y$. What is some other way(s) to find the cumulative distribution of this joint distribution?

I like your solution. As you mentioned, $$P(U<V \textrm{ and } Z<X<Y)= P(Z<X<Y) = P(Z=\min(X,Y,Z))P(X<Y) = \frac{1}{3}\cdot \frac{1}{2}$$
because the event $Z<X<Y$ is contained in the event $U<V$.
Now, you can consider all 6 cases,
$$P(U<V \textrm{ and } Z<X<Y)$$
$$P(U<V \textrm{ and } X<Z<Y)$$
$$P(V<U \textrm{ and } Z<Y<X)$$
$$P(V<U \textrm{ and } Y<Z<X)$$
$$P(U<V \textrm{ and } X<Y<Z)$$
$$P(U<V \textrm{ and } Y<X<Z)$$
Does that make sense?
Here is another idea:
\begin{align}
&P(U<u, V<v, z<Z<z+dz) \\& =P(U<u, V<v |z<Z<z+dz) P(z<Z<z+dz) =\\
 &=P(U<u |z<Z<z+dz) P(V<v|z<Z<z+dz) P(z<Z<z+dz)  \end{align}
Therefore,
\begin{align}
&P(U<u, V<v, z<Z<z+dz) \\& =P(X<u, Z<u |z<Z<z+dz) P(Y<v, Z<v |z<Z<z+dz)  \exp(-z)dz\end{align}
If $u>z$ and $v>z$
\begin{align}
&P(U<u, V<v, z<Z<z+dz) \\& =(1-\exp(-u) )(1-\exp(-v) ) \exp(-z)dz\end{align}
Do the other cases follow similarly?
