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 $E[XY]/E[Y^2]$

Intuitively, take the singular density in $\mathbb{R}^2$ that is only nonzero along some line $X = a Y$, $var(Y) = b$, $Y$ always nonzero, and that has $E[X] = E[Y] = 0$, as you required. Then:
$$
E\left[\frac{X}{Y}\right] = E[a] = a
$$
and
$$
E[XY] = a\,E[Y^2] = a \; var(Y) = ab.
$$
So you cannot compute $E\left[\frac{X}{Y}\right]$ from the covariance alone.
I don't think that there are easily obtainable information which, would you avail yourself of them in addition to $E[XY]$, were to give you $E[X/Y]$. But you can of course always construct some (silly) additional information satisfying your request, e.g. the knowledge of $E[X/Y - XY]$ (just kidding).
