Let $X$ and $Y$ be two independent $\text{Expo}(1)$ random variables. Let $M:=\max(X,Y)$ and $L:=\min(X,Y)$. How do I show that $M-L\sim \text{Expo}(1)$?

I have made an attempt as follows:

We note that $L\sim\text{Expo}(2)$. By the continuous version of the law of total probability we have

$$P(M-L\le t)=\int_{0}^{\infty}P(M-L\le t|L=s)\;f_L(s)\;\mathrm{d}s\\=\int_{0}^{\infty}P(M\le t+s|L=s)\;f_L(s)\;\mathrm{d}s$$

But I am stuck here, and I can't continue. How do I continue from this, or is there any other way to show that $M-L\sim \text{Expo}(1)$?

  • 2
    $\begingroup$ You need the joint distribution of $(M,L)$ [easy] and then derive the marginal distribution of $M-L$ from the joint, using convolution [easy]. $\endgroup$ – Xi'an Apr 14 '18 at 14:50
  • $\begingroup$ @Xi'an The author has not introduced convolutions yet. So is there a way to get around this without using convolutions? $\endgroup$ – Supreeth Narasimhaswamy Apr 14 '18 at 14:55
  • $\begingroup$ If you do not use the CDF method you can use a change of variables $U=M-L$ and $V=L$. But you need the joint density of $(M,L)$ first, which can be derived using order statistics or otherwise. $\endgroup$ – StubbornAtom Apr 14 '18 at 15:02
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    $\begingroup$ Here is a relevant post on math.se which uses the memoryless property of exponential distribution. In your case $n=2$. $\endgroup$ – StubbornAtom Apr 14 '18 at 15:17
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    $\begingroup$ What do you mean by "the author"? If you are referring to a textbook or a course please provide a reference. And add the self-study tag. $\endgroup$ – Xi'an Apr 14 '18 at 16:57

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