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How do you generate two $N(0, 1)$ standard normal distribution random variables with correlation $\rho$ if you have a radom number generator for standard normal distribution.

The solutions starts off by saying "Two $N(0, 1)$ randomv ariables $x_1, x_2$ with a correlation $\rho$ can be generated from indepnedent $N(0, 1)$ random variables $z_1, z_2$ using the following equations

\begin{align} x_1 &= z_1\\ x_2 &= \rho z_1 + \sqrt{1 - \rho^2}z_2" \end{align}

I know that the correlation coefficient has the formula $\rho = \frac{Cov(x, y)}{\sqrt{Var(x) Var(y)}}$, but I dont know how to deduce the two equations above from this definition.

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  • $\begingroup$ I would try the naive approach. The second r.v. has to be some function of the first, so let's try $x_2 = ax_1 + b z_2$ and plug this in the correlation coef. formula $\endgroup$
    – thewatcher
    Oct 7, 2021 at 0:50

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If you take for granted that it will be in the form $az_1 + bz_2$ then the variances add so the variance is $a^2+b^2$ which needs to be $1$. So $b=\sqrt{1-a^2}$. Then the correlation coefficient is just the covariance since the variances are both $1$, and the covariance is obtained as

$$\mathrm{Cov}(x_1,x_2)=\mathrm{Cov}(z_1,az_1+\sqrt{1-a^2} z_2)=\mathrm{Cov}(z_1,az_1)+\mathrm{Cov}(z_1,\sqrt{1-a^2} z_2)=a+0=a$$

so $a=\rho$.

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