# Conditional distribution and correlation in multivariate normal distribution.

Consider the random vector $(X_1, X_2, X_3)^T \sim \mathcal{N}_3 (\underline{\mu}, C)$, where $\underline{\mu}=(\mu_1, \mu_2, \mu_3)^T$ and

$$C=\begin{bmatrix} 1 & \rho & \rho^2 \\ \rho & 1\ & 0\\ \rho^2 & 0 & 1 \end{bmatrix}$$

$(0 < \rho < 1)$ is a given parameter. Determine the conditional distribution of $(X_1, X_2)^T$ conditioned on $X_3=x_3$.

Now suppose, that $\mu_1=\mu_2=\mu_3=0$, so $(X_1, X_2, X_3)^T \sim \mathcal{N}_3 (\underline{0}, C)$ and $C$ remains the same.

Find the multiple correlation between $(X_1, X_2)^T$ and $X_3$.

Find the partial correlation between $X_1$ and $X_2$, after eliminating the effect of $X_3$.

I am familiar with the definition of conditional distribution and correlation, but I haven't really solved any exercises in multivariate cases. Anything which can help me to get started is appreciated.

• I'm assuming all instances of ! are indeed exclamation marks rather than factorial notation. Please remove these to avoid ambiguity. – Jason Oct 3 '17 at 18:25
• Hopefully it won't cause confusion now :) – Atvin Oct 3 '17 at 18:26