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.)
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