# Calculate E[X/Y] from E[XY] for two random variables with zero mean


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.

• Thank you for your link to Ratio distribution. I should be specifically in the case of Correlated central normal ratio Feb 11, 2022 at 12:43

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).