# Distributions of random variables dependent on exponential random variables

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

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...

• Yep that was what I was thinking along those lines. Z & O are for a different question but if it were for Z rather than W would it just be the same pdf or do the max/min matter?
– user672283
Feb 21, 2021 at 12:17
• @m4thb0y : max pdf is different..it is not difficult to calculate it Feb 21, 2021 at 12:19
• Just do $P(Z<z)=P(X<z,Y<Z)$?
– user672283
Feb 21, 2021 at 12:23
• @m4thb0y starting from $F_Z=P(X<z,Y<z)$ you will realize that $F_Z=[F_X(z)]^2$ Feb 21, 2021 at 12:25