# How can we generate pairs of correlated random numbers?

If I can generate normal random numbers in $N(0,1)$, how can I generate two dependent random numbers, $Y_1$ and $Y_2$ with means $\mu_1$, $\mu_2$ and $\sigma_1$, $\sigma_2$ and correlation coefficient $\rho$ respectively? Is there a known method for this problem?

## 1 Answer

If $X_1$ and $X_2$ are i.i.d. standard gaussian, then $X_3=\rho X_1+\sqrt{1-\rho^2}X_2$ is standard gaussian as well and the correlation of $X_1$ and $X_3$ is $\rho$. Hence, starting from $X_1$ and $X_2$, one can use $$Y_1=\mu_1+\sigma_1X_1,\quad Y_2=\mu_2+\sigma_2\left(\rho X_1+\sqrt{1-\rho^2}X_2\right).$$

• Thanks! Just a quick question, how can we prove that the correlation of $X_1$ and $X_3$ is $\rho$ then? – LBJ Feb 29 '12 at 21:11
• Since $X_1$ and $X_3$ are centered and have unit variance, their correlation is $E(X_1X_3)$, which one computes by expanding the product. – Did Feb 29 '12 at 21:21
• Got it again...I'm so stupid... – LBJ Feb 29 '12 at 21:22