I have a question about the following from Introduction to Probability by Blitzstein:
I was able to show $L \sim$Expo(2) and use $M-L= \vert X-Y \vert$ to perform a double integral to show $M-L \sim$Expo(1), but got stuck on showing $M-L,L$ are independent. I didn't use the memoryless property in the 1st 2 parts so I suspect it will come in for this last part. So in my current approach I was forced to show all 3 statements separately, so I was also wondering if there's a way to do it all at once, of it at least 2 statements can be shown simultaneously. I also want to add I asked some friends who know probability better than I do, and they suggested order statistics. But order statistics have not come up yet in this point of Blitzstein's book.
Let's think about how $M$ is distributed conditionally on $L=l$. We know that there was another exponential variable $L=l$ that it is greater than, but $X$ and $Y$ are independent, so it will be conditionally distributed like $X$ given $X>l.$ So we can write $$ P(M>m|L=l) = P(X>m|X>l).$$
Next let's look at the distribution of $Z=M-L$ conditional on $L=l.$ We have $$P(Z>z|L=l) = P(M>z+l|L=l) = P(X>z+l|X>l) = P(X>z)$$ where the last equality used the memoryless property. Since $P(Z>z|L=l)$ does not depend on $l$, $Z$ and $L$ are independent. We have $P(X>z) = e^{-z}$ since $X$ is a standard exponential, so $Z$ is a standard exponential.