TITLE: When can i get away with approximating the expected value of a ratio as the ratio of expected values
QUESTION [8 upvotes]: I'm actually an engineering student so I'm not too good with probability and was hoping someone may be able to help with the following:
So I have a ratio of discrete random variables. I want to be able to know when I can get away with approximating its expected value as a ratio of expected values. For example, for the continuous case I came across the following formulation from a Taylor series expansion
$\text{E}[x/y]=\text{E}[x]/ \text{E}[y]- \text{Cov}[x ,y]/ \text{E}[y]^2 + \text{E}[x] \text{Var}[y]/ \text{E}[y]^3$
Does this apply for discrete random variables? if not, what is its analogous expression?
thank you!
hadi

REPLY [3 votes]: A  simple, general, good approximation purely in terms of expectations does not seem achievable.  For example, let $\epsilon$ be a very small positive number, and let $X$ and $Y$ be independent, each taking on the value $\epsilon$ with probability $1/2$, and $1$ with probability $1/2$.  Then $E(X/Y)$ is very large, but the various moments are all quite reasonable. 
Possibly some reasonable approximations are possible if all values of $Y$ are well away from $0$.

REPLY [2 votes]: That last term doesn't make sense to me.  Let $E[x] = \mu_x$ and $E[y] = \mu_y$.
From the Taylor series of $x/t$ around $t=\mu_y$, $x/y = x/\mu_y - x(y -\mu_y)/\mu_y^2 + x (y - \mu_y)^2/\mu_y^3 + \ldots $ resulting in 
$E[x/y] = \mu_x/\mu_y - {\rm Cov}[x,y]/\mu_y^2 + E[x (y - \mu_y)^2]/\mu_y^3 + \ldots$, but $E[x (y - \mu_y)^2] \ne \mu_x {\rm Var}[y]$ unless $x$ and $(y - \mu_y)^2$ are uncorrelated.
In any case, it has nothing to do with discrete vs. continuous: it is just as valid for discrete random variables as it is for continuous ones.  The main point is that to have a good approximation (apart from that last problematic term) you'll want 
$y - E[y]$ to be (with high probability) small compared to $|E[y]|$.  You especially need to avoid $y$ near 0, which could make $E[x/y]$ be undefined.