Distributions of random variables dependent on exponential random variables

Question

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

Answer 1

Starting from

$$P(W>w)=P(X>w,Y>w)$$

you will easy realize that

$$W\sim \exp(2\lambda)$$

M is clearly a bernulli with parameter 0.5

I do not know what $Z$ and $O$ are given for...