# Simulating Correlated Normal Random Variables Given Uniforms

I want to simulate $3$ Normal random variables given their expectations, variances, correlations and three independent uniform $(0,1)$ observations.

Is my method correct?

First, produce three independent Normal $(0,1)$ random variables using the Inverse Transformation i.e. $X_i=\Phi^{-1}(U_i)$ where $U_i$ is the given uniform variable, $i=1,2,3$. We note that the popular methods for simulating Normal random variables will not work here because we are given fixed values of the uniforms.

Second, Find a Cholesky Decomposition $U$ such that $UU'=C$ where $C$ is the variance-covariance matrix of $X_1,X_2,X_3$.

Third, Our expectation vector being $\mu$, our required Normal vector being $Z$ and our original Normal vector $(X_1,X_2,X_3)$ being $X$, we can apply $Z=\mu+UX$ to get our desired vector.

• Yes this is correct. And standard. – Xi'an Apr 8 '15 at 8:47