Analytical approaches to approximate probability density functions of multivariate random functions

Question

Given a random multivariate function $f(x, y, z)$, where $x, y, z$ are independent and identically distributed random variables with a probability distribution $\rho(X)$, I aim to approximate the value of the probability density function of the function $f(x, y, z)$ at $f_0$. Besides numerical sampling methods, are there any other analytical methods to solve this problem?

Answer 1

The probability density $p(f_0 )$ is given by \begin{eqnarray} p(f_0 ) &=& \int dx\,\rho (x) \int dy\,\rho (y) \int dz\,\rho (z) \ \delta (f_0 - f(x,y,z)) \\ &=& \int dx\,\rho (x) \int dy\,\rho (y) \sum_{z_0 (x,y)} \frac{\rho (z_0 (x,y))}{|f^{\prime } (x,y,z_0 (x,y))|} \end{eqnarray} where the sum runs over all zeros $z_0 (x,y)$ of $f_0 - f(x,y,z)$ as a function of $z$ for given $x,y$, and $f^{\prime } $ denotes the partial derivative with respect to $z$. For simple enough $f$ (and $\rho $), you may be able to perform this integral analytically. Alternatively, you are left with a two-dimensional integral to be performed numerically, where the evaluation of the integrand involves searching for the zeros $z_0 (x,y)$ and calculating the partial derivative $f^{\prime } $ there. Cases where $f^{\prime } $ becomes small or vanishes require special treatment (such as, e.g., exchanging the roles of $x,y,z$).