# Generate jointly distributed random coefficients with given mean and variance from $N(0,1)$ and some matrix $L$

Suppose $$\beta_1 \sim N(m_1,s_1^2)$$, $$\beta_2 \sim N(m_2,s_2^2)$$ and $$cov(\beta_1,\beta_2) = s_{12}$$. Now generate draws of these random coefficients from draws of two independent standard normal distributed variables $$z_1$$ and $$z_2$$. Give the $$2\times 2$$ matrix $$L$$ that satisfies: $$\begin{bmatrix}\beta_1\\\beta_2\end{bmatrix} \sim \begin{bmatrix}m_1\\m_2\end{bmatrix} + L \begin{bmatrix}z_1\\z_2\end{bmatrix}$$

Hint: make a guess and verify that L is Cholesky decomposition of $$\Sigma = \begin{bmatrix} s_1^2 & s_{12} \\ s_{12} & s_2^2\end{bmatrix}$$ by computing $$LL'$$.

What I tried is to substitute $$L = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ carry out the matrix multiplication and addition to get: $$\begin{bmatrix}\beta_1\\\beta_2\end{bmatrix} \sim \begin{bmatrix} m_1 + az_1 + bz_2\\ m_2 + cz_1 + dz_2 \end{bmatrix}$$

My understanding is that this means that $$\beta_1$$ is distributed as $$m_1 + az_1 + bz_2$$ and $$\beta_2$$ as $$m_2 + cz_1 + dz_2$$. Is this correct?

My next step is to find the expected values, variances and covariance of the RHS and set them equal to the given moments for $$\beta_1$$ and $$\beta_2$$. This gives me the following system of non-linear equations: \begin{align} a^2 + b^2 & = s_1^2 \\ c^2 + d^2 &= s_2^2\\ ac+bd &= s_{12} \end{align}

But how to continue from here? Set for example $$a=1$$ or $$a=0$$ and try to solve the system? Or am I thinking in the wrong direction? Thanks.

• $L$ is lower triangular, allowing you to solve for $a$; that lets you solve for $c$, and then for $d$ – Glen_b Feb 6 at 1:58

I think you used the Cholesky decomposition where $$L$$ is a lower triangular matrix, meaning $$b=0$$. I hope this helps.