Subdivide Z into $X_1,X_2$, s.t. $Z=X_1+X_2$ and ($X_1,X_2$) obey bivariate normal

Question

For a simulation in a research project, I am trying to randomly "appropriate" (meaning subdivide into two components) known values of $Z$ into $X_1$ and $X_2$ such that $Z=X_1+X_2$ in a way that respects their known bivariate normal distribution ($X_1,X_2 \sim N(\mu,\Sigma)$). Is there a way to do this directly?

Not knowing a direct way, I imagined I could compute a distribution of a new variable $U$ that is the conditional distribution given $X_1+X_2=c$, where $c$ is some constant. That is $U\mid(X_1+X_2=c)$, then "appropriate" $Z$ into $X_1$,$X_2$ depending on the realization of $U\mid(Z)$. Perhaps this is equivalent to centering the axis at $\mu$ then rotating the axis 45 degrees?

Once I know the theory, I will need to implement this in R.

Answer 1

When $(X_1,X_2)$ enjoy a bivariate normal distribution, $Z = X_1+X_2$ is a normal random variable also, with mean $\mu_Z = \mu_1+\mu_2$ and variance $\sigma_Z^2 = \Sigma_{11} + \Sigma_{22} + 2\Sigma_{12}$. So, to simulate $Z$, all you need to ask R to do is generate samples of a $N(\mu_Z,\sigma_Z^2)$ random variable.

If you wish to determine the conditional distribution of $(X_1,X_2)$ given $Z$, then be aware that the conditional distribution is not a bivariate normal distribution in the usual sense of the bivariate normal joint density. $X_1$ and $X_2$ indeed are bivariate normal random variables but their covariance matrix $\Sigma$ is singular and they don't have a two-dimensional joint density. The distribution is degenerate in that given the value of $Z$ is $c$, the value of $X_2$ is necessarily $c-X_1$. So, to simulate the conditional distribution given $Z = c$, you can simulate the value of $X_1$ given $Z=c$ and then set $X_2$ equal to $c-X_1$. Note that $(X_1, X_1+X_2)$ is bivariate normal with mean $(\mu_1, \mu_1+\mu_2)$ and covariance matrix $$\begin{bmatrix} \sigma_1^2 & \sigma_1^2 + \Sigma_{12}\\ \sigma_1^2 + \Sigma_{12}& \sigma_Z^2\end{bmatrix}.$$ I will leave it as an exercise for you to determine the conditional distribution of $X_1$ given $X_1+X_2 =c$ from this information. Or, if you are feeling lazy, see this answer for ideas on how to proceed.