Calculate E[X/Y] from E[XY] for two random variables with zero mean I have two random variables $X$ and $Y$, both with zero mean.
$\newcommand{\E}{\mathrm{E}}$
$\newcommand{\Var}{\mathrm{Var}}$
$\newcommand{\Cov}{\mathrm{Cov}}$
Let's suppose I only know their covariance, which is, in this case, simply $\mathrm{E}[XY]$.
Can I easily calculate $\mathrm{E}\left[\frac{X}{Y}\right]$ from $\mathrm{E}[XY]$?
If not, what other information would I need to calculate $\mathrm{E}\left[\frac{X}{Y}\right]$?
EDIT: I add some assumptions: $X$ and $Y$ are Gaussian and their covariance is $\neq 0$.
Thus, referring to @j-delaney 's answer, I should be in the case of Correlated central normal ratio.
The Correlated central normal ratio is a Cauchy distribution for which the mean is not defined (thus $\mathrm{E}\left[\frac{X}{Y}\right]$ is not defined). The $x_0$ parameter of the Cauchy distribution, in my specific case, should be $E[XY]/E[Y^2]$

You will have to know the full joint distribution of $X$ and $Y$ in order to calculate $$E[X/Y] = \int (x/y) p(x,y) ~dx dy. $$
Note that $E[X/Y]$ might not even be defined - this is the case for example when $X$ and $Y$ are normally distributed, and the ratio has a Cauchy distribution which has no mean.
See also Ratio distribution.
