probability - Show that $P(X

Let $X$ and $Y$ be nonnegative, independent continuous random
variables.

(a) Show that $$P(X (b) What does this become if $X ∼ \exp(λ_1)$ and $Y ∼ \exp(λ_2)$?

I don't understand what exactly $P(X

Answer

Your interpretation of $\mathbb P(X < Y)$ is correct -- the next step is turning that idea into a mathematical expression you can work with.

If you have two random variables with joint density $f(x,y)$, then $\mathbb P(X < Y)$ can be computed as
$$\mathbb P(X < Y) = \iint_{\{x < y\}} f(x, y) \, \textrm d x \, \textrm d y.$$
Since these variables are supported on $[0, \infty)$, one could also write this as
$$\int_0^{\infty} \int_0^y f(x, y) \, \textrm dx \, \textrm dy.$$
But in the case where $X, Y$ are independent, the joint density $f(x,y)$ is just a product of their individual densities. Can you take it from here? (I can provide more hints if not.)