Let $X \sim \text{exp}(\lambda), Y \sim \text{exp}(\mu), Z \sim \exp(\gamma)$. Let U = min(X, Y), V = min(Y, Z). Compute $P(U > u, V > v)$ where u, v > $0$. (Consider two cases: one when $u < v$ and the other when $ u > v$). Differentiate what you find with respect to u and v to find the joint density function (its integral will not equal 1).
$X, Y, Z$ are independent, by the way.
I found $U \sim \exp(\lambda + \mu)$ and $V \sim \exp(\mu + \gamma)$ by doing $P(U > x)$ and $P(V > y)$. But I suppose I am not so sure how to continue. Maybe please someone can help me. I am struggling because $\mu$ is a shared variable
I'm assuming $X,Y,Z$ are independent.
If $u \le v$ then $$P(U > u, V > v) = P(X > u, Y > v, Z > v) = P(X>u) P(Y>v) P(Z > v) = e^{-\lambda u - \mu v - \gamma v}.$$ The other case $u \ge v$ is similar.