Let underline Xsim N3(underlinemu,Sigma) with Sigma=(beginsmallmatrix1&rho&rho^2rho&1&0rho^2&0&1 endsmallmatrix). Then the conditional distribution of (X1,X2) given X3=x3 is N2(underlinemu^,Sigma^) where underlinemu^ and Sigma^ are respectively given by:

(C) binommu1+rho^2(x3-mu3)mu2, (beginsmallmatrix1-rho^4&rhorho&1endsmallmatrix)

Q159Section III (Statistics)159 of 190

The standard multivariate-normal conditional mean and covariance formulas

Let $\underline X\sim N_3(\underline\mu,\Sigma)$ with $\Sigma=\left(\begin{smallmatrix}1&\rho&\rho^2\\\rho&1&0\\\rho^2&0&1 \end{smallmatrix}\right)$ . Then the conditional distribution of $(X_1,X_2)$ given $X_3=x_3$ is $N_2(\underline\mu^*,\Sigma^*)$ where $\underline\mu^*$ and $\Sigma^*$ are respectively given by:

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