# Mean and variance of non-linear function of multiple Gaussian distributed variables

Given random variable vector $X=[X_1,...,X_n]$, where $X_i \sim N(\mu_i,\sigma_i^2)$ and non linear function $f(X) \in \Re^1$, is there a generalized method for finding $E(f),Var(f)$?

Specifically, I am interested in finding $var(f)$ where $$f([\theta_1, \theta_2, \theta_3]) = -\cos\left(\theta_{1}\right)\,\left(\cos\left(\theta_{2}\right)-\cos\left(\theta_{2}-\theta_{3}\right)\right)$$

Is this analytically determinable? Or would a monte-carlo based approximation be more appropriate?

Thanks!

• You can use the multivariate delta method. – Joda Sep 3 '18 at 4:53