# Joint cdf and pdf of the max and min of independent exponential RVs [duplicate]

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$.

Define two new random variables by

$W = \min({X,Y})$

$Z = \max({X,Y})$

Find the joint cdf of $W$ and $Z$, and use it to find their joint pdf.

I know I need to start out by splitting up the event $\{W \le w, Z \le z\}$ as the union of $\{W \le w, Z \le z, X \le Y \}$ and $\{W \le w, Z \le z, X > Y \}$

## marked as duplicate by Chill2Macht, Joey Zou, naslundx, Claude Leibovici, user91500Aug 15 '16 at 7:00

Hint: Note that $$\{Z \leq t\} = \left[\{Z\leq t\}\cap \{W \leq s\}\right]\cup\left[\{Z\leq t\} \cap \{W > s\}\right]$$ Therefore you will find that $$P(Z \leq t, W \leq s) = P(Z \leq t) - P(Z \leq t, W > s)$$ where the right side is a reasonable calculation. For example $$P(Z \leq t, W > s) = P(X \leq t, Y \leq t, X > s, Y > s) = P(s < X \leq t, s < Y \leq t) =…$$