Expectation of the ratio of two $\mathcal{X}^2(1)$ random variables

Question

Let $X$ and $Y$ be two possibly dependent Chi-squared random variables both with $k=1$. Is it possible for $\mathbb{E}[X/Y]$ to be unbounded?

Answer 1

A Chi-squared random distribution with $k=1$ can be realized as the square of a standard normal random variable. So take $X = U^2$ and $Y = V^2$, where $U$ and $V$ are independent $N(0,1)$. Then $U/V$ has a Cauchy distribution, which already has no expected value (i.e $\mathbb E[|U/V|] = \infty$). $U^2/V^2$ is even worse, as $U^2/V^2 \ge |U/V| - 1/4$.

Answer 2

Yes. For instance,

$$\mathbb{E} 1/Y=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{1}{x^2} e^{-x^2/2}\textrm{d}x=\infty $$ So if $X$ and $Y$ are independent, we have $\mathbb{E}(X/Y)=\infty$.