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


1 Answer 1


Starting from


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...

  • $\begingroup$ 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? $\endgroup$
    – user672283
    Feb 21, 2021 at 12:17
  • 1
    $\begingroup$ @m4thb0y : max pdf is different..it is not difficult to calculate it $\endgroup$
    – tommik
    Feb 21, 2021 at 12:19
  • $\begingroup$ Just do $P(Z<z)=P(X<z,Y<Z)$? $\endgroup$
    – user672283
    Feb 21, 2021 at 12:23
  • $\begingroup$ @m4thb0y starting from $F_Z=P(X<z,Y<z)$ you will realize that $F_Z=[F_X(z)]^2$ $\endgroup$
    – tommik
    Feb 21, 2021 at 12:25

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