## Finding the distribution of a function of $n$ random, normally distributed correlated variables

Given a random vector $X$ of $n$ normally distributed random variables, and an $n \times n$ covariance matrix of those variables with non-zero correlation terms, what is the generalized methodology to find the distribution of a non-linear function $f(X_1,X_2,\ldots,X_n)$ of the random variables of $X$?

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 You haven't given enough information; in particular, you haven't said that they are jointly normally distributed. That is the same as saying they are so distributed that every constant linear combination of them is normally distributed. If that is the case, and if the covariance matrix $V$ is nonsingular, then the density function is $\text{constant}\cdot \exp\left(\frac{-1}{2} (x-\mu)^T V^{-1}(x-\mu) \right)$. The "constant" is the reciprocal of $\sqrt(2\pi\det V)$. However, this is standard textbook stuff (google "multivariate normal distribution"); so maybe not right for this forum. – Michael Hardy Sep 6 2011 at 17:12 ....on the other hand, if you did not intend them to be multivariate normal, then there might be something in it that is appropriate for this forum, but then your question becomes rather uncomfortably vague until you say more than you have. (BTW, I intended $\sqrt{2\pi\det V}$, but the software here won't let me edit to correct typos in my comments. At stackexchange the software is somewhat more sophisticated.) – Michael Hardy Sep 6 2011 at 17:14 ....in other words: They could each be normally distributed, and correlated, but not jointly normally distributed, so that, for example, although $X_1$ and $X_2$ separately are normally distributed, $X_1+X_2$ might not be. That's the situation in which your question would involve something that might not be in textbooks. – Michael Hardy Sep 6 2011 at 20:51 @jaha, here is an exercise: for any distribution $\mu$, there exists a function $f$ such that the distribution of $f(X_1)$ is $\mu$. – Didier Piau Sep 6 2011 at 21:18 @Didier: Yes, but the question could still be construed as meaning: What does information about $f$ tell us about the probability distribution of $f(X_1)$? It's not a very clearly written question as it stands. My suspicion is that most likely he has something fairly specific in mind that he hasn't made clear. – Michael Hardy Sep 6 2011 at 23:36