What is $E(X/Y)$ if $X$ and $Y$ are independent standard normal?

Question

Let $X \sim \mathcal{N}(0,1)$ and $Y \sim \mathcal{N}(0,1)$ independent from $X$. I know that $E(\frac{X}{Y})=E(X)E(\frac{1}{Y})$; the first term is zero and the second infinity? But do not know how to proceed from there, or even if it the expected value exists.

Answer 1

Hint: by independence, $$\mathbb{E}\left|\frac{X}{Y}\right| = E|X| \cdot E\left[\frac{1}{|Y|}\right].$$ Prove that $E\left[\frac{1}{|Y|}\right] = \infty$ (you can write down an appropriate integral and compare it to $\int_{-1}^{1} \frac{1}{|y|}\,dy$) and that $E|X| > 0$.