First, I need to sample 10,000 vectors $Y_{i}=\begin{bmatrix}Y_{1i} & Y_{2i} & Y_{3i}\end{bmatrix}^{T}$ that are multivariate normal $N(\mu,\Sigma)$ with $\mu = \textbf{0}$ and $$\Sigma = \begin{bmatrix}1 & 4.5 & 9\\ 4.5 & 25 & 49\\ 9 & 49 & 100\end{bmatrix}.$$ I don't need help here. I know that you can draw iid standard normal vecotrs $Z_{i} = \begin{bmatrix}Z_{1i} & Z_{2i} & Z_{3i}\end{bmatrix}^{T}$ and set $Y_{i} = CZ_{i}$ where $C = \Sigma^{\frac{1}{2}}$.
My question is, how do you sample 10,000 variables $Y_{1i}$ from the distribution $P(Y_{1i} \mid Y_{2i} = Y_{3i} = 1)$?
I'm drawing the samples in R, but I'm more interested in the theory than the code. I don't know how to approach this.
Given that $Y_{i}=\begin{bmatrix}Y_{1i} & Y_{2i} & Y_{3i}\end{bmatrix}^{T}$ has a multivariate distribution, $Y_{1i}$ is also distributed normally (this is by definition).
Now, when you condition on $Y_{2i}=Y_{3i}=1$, you can think of it as if you have "observed" the values for $Y_{2i}$ and $Y_{3i}$ and you just take into account that they are now static in value when looking at the probability distribution of $Y_{1i}$.
To think of it geometrically I think it's easier explained in a 3-d space. Imagine $X=(Z_1,Z_2)$ is bivariate normal. Suppose each $Z_i$ has a standard normal distribution. They will have the following graph (ignore the values on the axes):
Imagine now that you "observe" that $Z_1=0.5$. You now fix $Z_1$ at $0.5$ and slice the graph so you see what values does the density take when you consider the values of $Z_2$. Here's how this would approximately look like (ignore the values on the axes):
This picture shows what's the distribution of both of the variables, but the idea is the same.