I am given a dataset with in each column a set of data pertaining to a different random variable. I know that the data are normally distributed. Now how can I find the estimated mean and variance of the last observation from the first random variable conditional on the last observation of the other two random variables?
I thought of finding the conditional pdf and the calculating the maximum likelihood estimator. However, I am not sure how I can condition on two random variables. Could anyone help me in the right direction?
Thank you in advance for your help.
Best Answer
Ok. $\Sigma$ is the $3\times 3$ covariance matrix of the joint distribution of your three variables (just look at the article). It is then unequally partitioned into sub-matrices. Denoting $v_{ij}$ the elements of $\Sigma$, we have, following the notation of the article,
$$\Sigma_{11} = v_{11}=\sigma^2_1\;,\; \Sigma_{12} = [v_{12}\;\; v_{13}]\;,\; \Sigma_{21} = \left[ \begin{matrix} v_{21}\\ \\ v_{13}\end{matrix} \right]$$ $$\Sigma_{22} =\left[\begin{matrix} v_{22} &v_{23} \\ v_{32} & v_{33} \end{matrix}\right] = \left[\begin{matrix} \sigma^2_{2} &v_{23} \\ v_{32} & \sigma^2_{3} \end{matrix}\right]$$
Then the conditional expectation function $E(X_1\mid X_2,X_3)$ is
$$E(X_1\mid X_2,X_3) = \mu_1 + \Sigma_{12}\Sigma^{-1}_{22}\left [\begin{matrix} X_2-\mu_2 \\ \\X_3-\mu_3\end{matrix} \right] $$
I guess now you can work the term for the conditional variance.