How to generate a multivariate normal distribution by conditioning on the normally distributed mean of each sample?

Question

I would like to generate a joint distribution of random variables $X$ and $Y$ from a multivariate normal distribution with mean zero and symmetric covariance matrix. However, each generated $x_i$ and $y_i$ needs to be constrained by a mean, $z_i$, that was previously generated from a normal distribution with the same variance as $X$ and $Y$. For each sample generated, how do I condition on $z_i$ to sample $x_i$ and $y_i$, while maintaining the desired covariance matrix for the joint distribution of $X$ and $Y$?

Here is an example: Let's say I ultimately want a multivariate normal distribution of $X$ and $Y$ with: $ \mu = \left(\matrix{0 \\ 0}\right)$ and $\Sigma = \left(\matrix{\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}}\right)$, where $\sigma_{11}=\sigma_{22}$. I have sampled $Z$ from normal distribution: $Z \sim \mathcal{N}(0,\sigma_{11})$. For each $z_i$, I want to sample $x_i$ and $y_i$ such that they have mean $z_i$, but overall will have a covariance matrix $\Sigma$ just described.

I think I can sample each $x_i$ and $y_i$ independently from a normal distribution with mean $z_i$, but I'm not sure what the variance should be. I have found a distribution to sample one value conditional on knowing the other that it covaries with, but I believe that's different from what I'm trying to do here. Thank you in advance for helping me identify the right distribution around each $z_i$!

Answer 1

If$$\left(\matrix{X \\ Y}\right)\sim \mathcal N \left(\mu,\Sigma\right)$$then $$X\sim\mathcal N(0,\sigma_{11})$$and$$Y|X\sim\mathcal N(\sigma_{12} X/\sigma_{11},\sigma_{22}-\sigma_{11}^2/\sigma_{11}^2)$$ meaning that $Z$ and $X$ have the same distribution.