Let $X$ and $Y$ be two independent random variables such that $X,Y\sim Exp(\lambda)$ Define:
$$W=\min(X,Y)$$ $$Z=\max(X,Y)$$ $$O=Z-W$$ $$M=\mathbf1_{X\le Y}=\begin{Bmatrix} 1 & \text{ if } X\le Y \\ 0 & \text{ if } X\gt Y \end{Bmatrix}$$
Identify the distributions for $W$ and $M$ (i.e. for $w \ge 0$, compute $P(W>w)$ and $P(M=1)$).
So for this question I was just wondering whether I should compute the joint probability density function $P(X>x,Y>y)$ for $W$
If X and Y both follow exponential distributions would it be reasonable to assume $M$ follows a uniform distribution with $P(M=1)=\frac{1}{2}$?
Any help would be appreciated as I am a bit stumped and feel I am missing something obvious