Let X and Y be independent exponential random variables with parameters λ > 0 and µ > 0, respectively. Let $U = max(X,Y)$ and $V = min(X,Y)$. Find the density function $f_{(U,V)}(u,v)$.
I've done this problem but I'm not confident in my answer.
I got $f_{(U,V)}(u,v) = λe^{(-λv- \mu u)}\mu + \mu e^{(-\mu v - λu)}λ - λe^{(-λv - \mu u)}\mu$
Could someone verify my answer or walk me through how they would solve it. I would really appreciate it! Thanks!
I would proceed by a limit argument. Let $a < b$ and let $h$ be such that $a + h < b$. Then \begin{align} P(V \in (a, a + h), U \in (b, b + h)) &= P(X \in (a, a + h), Y \in (b, b + h)) + P(Y \in (a, a + h), X \in (b, b + h)) \\ &= P(X \in (a, a + h))P(Y \in (b, b + h)) + P(Y \in (a, a + h))P(X \in (b, b + h)) \end{align} Thus by dividing both sides by $h^2$ and letting $h \to 0^{+}$ we get \begin{align} f_{V, U}(a, b) = f_{X}(a)f_{Y}(b) + f_{Y}(a)f_{X}(b). \end{align} This argument assumed $f_{V, U}$ would be continuous at $a, b$, but you can arrive at the same result without this assumption by computing $F_{V, U}(a, b)$ and then using $$f_{V, U}(a, b) = \frac{\partial}{\partial a \partial b}F_{V, U}(a, b).$$