Assume we have three random variables $x, y, z$, and they follow joint Gaussian distribution. We define $W$ as a new random variable, which is the function of $x, y, z$, denoted as $f(x, y, z)$. The question is how to obtain the probability density function of $W$ in the following two conditions:
1、$f$ is a closed-form function;
2、$f$ is not a closed-form and the value of $f(x, y, z)$ can be determined by simulation methods such as Monte Carlo simulation when $x, y, z$ are fixed.
The question is how to obtain the density function of $W$ in the above two conditions. Does such a method exist? Any comment will be thankful.
These settings are quite general. In the case of 1, it might be possible to calculate $F_W(x) = P(f(X,Y,Z) \le x) $, and differentiate it to get the density function, but this could be tough if $f$ is not easy to work with. In the setting 2, it is at least easy to get an approximation of the density if one is able to simulate the random variable $W=f(X,Y,Z)$. In this case one would simply simulate many, many replicates of $W$, and look at the histogram. One could also apply proper density estimation techniques to "smooth" the histogram. See e.g.
https://machinelearningmastery.com/probability-density-estimation/