I’m trying to construct correlated bivariate normal random variables from three univariate normal random variables. I realize there is a formula for constructing a bivariate normal random variable from two univariate random normal variables, but I have reasons for wanting to adjust two previously sampled variables by a third in order to give them a correlation $ \rho$ .

Based on the approaches for constructing bivariate normals from two univariate normals, I came up with the following approach and need help verifying its correctness.

First, imagine that we have three univariate normal variables. For simplicity here, we just assume they all have $ \sigma=1$ .

$ $ X_0 \sim Normal(0, 1)$ $ $ $ Y_0 \sim Normal(0, 1)$ $ $ $ Z \sim Normal(0, 1)$ $

Given these three univariate random variables, I construct two new random variables using the following linear combinations:

$ $ X = |\rho| * Z + \sqrt{1-\rho^2} * X_0$ $ $ $ Y = \rho * Z + \sqrt{1-\rho^2} * Y_0$ $

where $ \rho \in [-1, 1]$ represents the correlation coefficient between the two univariate normals.

Can someone help me formally verify that $ X$ and $ Y$ are now correlated random variables with correlation $ \rho$ ?

I’ve convinced my self through empirical simulation. Here are plots of values sampled from $ X$ and $ Y$ for the cases where $ \rho=0$ , $ \rho=1$ , and $ \rho=-1$ .

Plot of X vs. Y for rho of 0

Plot of X vs. Y for rho of 1

Plot of X vs. Y for rho of -1

These plots are exactly as I would expect, but it would be nice to have a formal proof based on my construction. Thanks in advance!