# probability re: comparing 2 iid exponential random variables

Let $$X, Y$$ be iid exponential random variables with parameter $$1$$. Then, what is the probability that $$X < Y + 1$$?

I know how to compute $$P(X < Y)$$ (from integrating the joint PDF, which is the same as the product of the marginal PDFs for X and Y because X and Y are independent, and the marginal PDF is the PDF of an exponential random variable with parameter $$1$$).

Maybe I'm missing something obvious, but the $$+ 1$$ is really tripping me up.

Given that you know how to integrate the joint density, then all you have to do is modify the region of integration suitably: The inequality $$X < Y + 1$$ with the added conditions $$X > 0$$, $$Y > 0$$, would represent what region of the $$(X,Y)$$ coordinate plane? Plot the equation $$Y = X - 1$$. What does that look like? Now, sketch the region satisfying the inequalities. Then consider how to set up an iterated integral over which you would integrate the joint density: one order of integration will be more inconvenient than the other, so choose carefully.