Transformation of exponential distributions

Question

$X\sim\operatorname{Exp}(a)$ and $Y\sim\operatorname{Exp}(b)$ are independent random variables. What are $P(X>Y)$ and $P(X>2Y)$?

I know the answers are $\frac a{a+b}$ and $\frac a{a+2b}$ respectively, but I don't know how to solve this. Somebody help me thanks!

Answer 1

The probabilities you want to know can be calculated as the integral of the joint distribution over a specific domain: $$ \iint_D f(x,y)dydx. $$ the difficulty is to find the domain you need. For the first question, a description of this domain would be $0\le x<\infty$ and $0\le y \le x$. Note that this is not the only way to describe this domain.

Answer 2

Comment. Here is a simulation of the second part in R statistical software, based on 30,000 $(X, Y)$ pairs. I used $X \sim Exp(rate = 1/5)$ and $Y \sim Exp(rate = 1/3)$ so that $E(X) = 5$ and $E(Y) = 3.$ The approximation of $P(X > 2Y)$ would be better with more points.

x = rexp(30000, 1/5);  y = rexp(30000, 1/3)
mean(x > 2*y)
## 0.4556667  # aprx P(X > 2Y) = 5/11

The main point of this demonstration is the figure below, in which the 13,670 points corresponding to the event $\{X > 2Y\}$ are shown in red. This may help you set the limits on your integrals. The density of the points near the origin indicates that the joint density function is relatively large there.