I asked this question on http://math.stackexchange.com/questions/377140/random-variables-related-through-nonlinear-system-of-equations, however I received no answer for a while so I'm posting it here:

Lets assume two groups of random variables X and Y (the dimensionality of them is not important). I know probability distribution of X, but not of Y. I also know that Y is a function of X and they are related through system of nonlinear equations e.g. $$ \left\\{\begin{matrix} Y_{1}^2+X_{2}=10\\\ Y_{2}^2+Y_{1}=X_{1} \end{matrix}\right. $$ Suppose for this particular example that we cannot find analytic solutions of $Y$ in terms of variables $X$. Is there any theory that would enable to extract some probabilistic information (e.g. expectation, variance) about $Y$ without first finding analytic solutions? I.e. if $X_{1}$ and $X_{2}$ are independent random variables with gamma distributions, what tools could be used to infere for example at least expectation and variance on $Y_{1}$ and $Y_{2}$ without solving system explicitly? (The example is just for illustratory purposes, the real system of equations are different.)