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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.)

Tomas
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  • Since there was not answer/comment on this question for a while, I moved it to mathoverflow.net

    http://mathoverflow.net/questions/129386/random-variables-related-through-nonlinear-system-of-equations

     – Tomas May 02 '13 at 08:08

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