r.v. Law of the min On a probability space $(\Omega, A, P)$, and given a r.v. $(X,Y)$ with values in $R^2$. If the law of $(X,Y)$ is $\lambda \mu e^{-\lambda x - \mu y } 1_{R^2_+} (x,y) dx dy$, what is the law of the min $(X,Y)$?

Here is an attempt to complete what Sasha started:
$\mathbb P(Z>z)=\mathbb P(X>z,Y>z)$
$=\mathbb P(X>z)\mathbb P(Y>z)$
$=\displaystyle \left(\int_{z}^{\infty}\lambda e^{-\lambda x}dx\right)\left(\int_{z}^{\infty}\mu e^{-\mu y}dy\right)$
$=\left[\left.-e^{-\lambda x}\right|_{z}^{\infty}\right]  \left[\left.-e^{-\mu y}\right|_{z}^{\infty}\right]$
$= e^{-(\lambda +\mu)z}$
And then taking the negative of the derivative w.r.t $z\Rightarrow$ $\displaystyle (\lambda+\mu)e^{-(\lambda+\mu)z}\mathbb 1_{\mathbb R}(z)dz$, an exponential distribution.
