I want to know if there are approximate solutions to the joint distribution of functions of random variables, where the variables can be only normal or lognormal. The functions would be using only addition, multiplication and division. for example $$ f(X_1,X_2,...,X_N)=X_1+\frac{X_2 X_3}{X_4}+... $$ I just have the intuition that given these constraints, we can have a faster way of solving or getting an approximate solutions without using the jacobian with the inverse, that is not practical at all when we have 6,7 variables added, multiplied and divided by each other.
So far, I am finding the pdf through monte carlo simulations in MATLAB, but I want to know whether there is a mid-way that offers some algebraic insight between a seemingly impossible analytical solution, and a completely numerical Monte-Carlo based solution. Below is an example of the two functions which distributions I want to approximate.
$$ M_u=A_sF_y d \frac{(1-(A_sFy))}{(1.7f_c bd)}\\ $$ $$ V_u=(\lambda/6) b d \sqrt{fc} +\frac{(A_{sv} F_{yv} d)}{s_p} $$ where $b, d ,f_c ,F_y,A_s,A_{sv}$ can be normal or lognormal random variables.Through monte carlo simulation, I know that the both $M_u$ and $V_u$ are bell-shaped. How can that help me? can I just approximate them as normal random variables?
In terms of background, I have a Bsc of electrical engineering, and I am trying to brush up on my random variables and stochastic processes background.
EDIT: all the variables are independent.
Also attached are simulations of both functions.
