How can we generate pairs of correlated random numbers?

Question

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?

Answer 1

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 $$ Y_1=\mu_1+\sigma_1X_1,\quad Y_2=\mu_2+\sigma_2\left(\rho X_1+\sqrt{1-\rho^2}X_2\right). $$