# Approximation of mean of a rational function of random variables

Let $$\xi_i$$ with $$i\in\{1,\dots,n\}$$ be iid random variables and let $$Q(x,y)$$ be a rational function.

I need to compute one $$x$$ that satisfies $$\frac{1}{n}\sum_{i=1}^n Q(x,\xi_i)=0.$$ This is a horrible equation for big $$n$$ because each denominator is different, but $$\xi_i$$'s have small variance, so there seems to exist a root close to each root of $$Q(x,E[\xi_i])$$.

Is there any way to approximate this mean? Is there some kind of expansion or some limit theorem that can be used to get an approximation of a root of this equation?