If you want to draw a N×1random vector x with a multivariate normal distribution with mean zero and N×N variance matrix Σ , then you do the following:
- Draw N independent and identically distributed N(0,1) random variables, and stack them up in a vector Z.
- Calculate the Cholesky decomposition of Σ (the CC’ form). If Z is a vector of length k of independent random variables with unit (or at least constant) standard deviation; and § is a correlation matrix with Cholesky decomposition S=CC′S, then CZ with have population correlation S.
- Multiply x=CZ.
- Now, x is distributed normal with mean zero and variance Σ.
Two Distributions Case
Population correlation. This is a simple matter in the bivariate case of taking independent random variables with the same standard deviation and creating a third variable from those two that has the required correlation with one of the two random variables. If X1 and X2 are independent standard normal variables, then ( Y = rX2+sqrt(1-rr)*X1 ) will have correlation r between Y and X2 .
Here’s an example in R:
n = 10 r = 0.8 x1 = rnorm(n) x2 = rnorm(n) y1 = rx2+sqrt(1-rr)*x1
This is a mash up from: