# Correlation between normal random variables

Suppose $$X$$, $$Y$$, and $$Z$$ are normal random variables with means and variances of $$\mu_X, \mu_Y, \mu_Z$$ and $$\sigma^2_X, \sigma^2_Y, \sigma^2_Z$$ respectively. $$X$$ and $$Y$$ have correlation $$\rho$$. $$X$$ and $$Z$$ are independent, and $$Y$$ and $$Z$$ are independent. Then what is the correlation of $$X$$ and $$Y-Z$$? I know that $$Y-Z$$ has distribution $$\mathcal{N}(\mu_Y - \mu_Z, \sigma^2_Y + \sigma^2_Z)$$.

$$corr(X, Y-Z) = \frac{cov(X,Y-Z)}{\sigma_X\sigma_{Y-Z}}$$. $$cov(X,Y-Z) = cov(X,Y) - cov(X,Z)$$.

$$corr(X,Y) = \rho = \frac{cov(X,Y)}{\sigma_X\sigma_Y}$$. So $$cov(X,Y) = \rho \sigma_X \sigma_Y$$. $$cov(X,Z) = 0$$ since they are independent. So $$cov(X,Y-Z) = cov(X,Y)$$.

So $$corr(X, Y-Z) = \frac{cov(X,Y)}{\sigma_X\sqrt{\sigma^2_Y + \sigma^2_Z}} = \frac{\rho \sigma_X \sigma_Y}{\sigma_X\sqrt{\sigma^2_Y + \sigma^2_Z}} = \frac{\rho \sigma_Y}{\sqrt{\sigma^2_Y + \sigma^2_Z}}$$.

Is this right?

• What is the covariance of $X$ and $Y$? What is the covariance of $X$ and $Y-Z$? What is the correlation of $X$ and $Y-Z$? Commented Dec 12, 2021 at 4:34

You have a joint (normal) distribution of $$(X,Y,Z)'$$, can you think of a transformation to get the joint distribution of $$(X,Y-Z)$$?
Hint: If you take A = $$[1,-1,0]$$ you get the distribution of $$(X-Y)'$$
• As @Glen points out, this approach requires an additional assumption. However, the idea is correct, because the question supplies full information about the variance-covariance matrix of $(X,Y,Z).$ Whether that has a multivariate Normal distribution is irrelevant.