Calculate $E(Y)$

Question

Let $(X, Y)$ be a randomly distributed vector with distribution $f_{X, Y}(x, y)=xe^{-x(1+y)}$ if $x, y \geq 0$ (and $0$ in other case). I have to calculate $E(Y)$. First of all I have calculated the marginal distribution of $Y$ which is: $$f_{Y}(y)=\int_0^{\infty}xe^{-x(1+y)}\: dx = \frac{1}{(1+y)^2}$$ Then $$E(Y)=\int_0^{\infty}\frac{y}{(1+y)^2}=\infty$$ As this diverges, what do I do? How do I calculate the expected value? Or it doesn't exist?

Answer 1

(Answering so it can be accepted, but making it a community wiki since I'm not contributing substantively beyond what Mostafa Ayaz already said:)

It's perfectly acceptable for a random variable to have an infinite expectation, even in cases when it is finite with probability $1$. There are somewhat artificial examples of this (such as the one in this problem), and then are important examples of this phenomenon as well (such as the time of return of a one-dimensional random walk to its origin).

Your work is correct and has led you to the right answer of $\infty$.