# How to solve for the Weighted Conditional Expectation $(\boldsymbol{x_1}|\boldsymbol{x_2}=a)$

I am working with Conditional Normal Distributions. In particular, I am attempting to find the conditional mean:

$$\boldsymbol{\bar \mu} = \boldsymbol\mu_1 + \boldsymbol\Sigma_{12} \boldsymbol\Sigma_{22}^{-1} \left(\mathbf a - \boldsymbol\mu_2\right)$$

I understand that there is a general formula for applying a weighted mean. I am attempting to do something similar with the conditional mean.

Assuming:

$$\boldsymbol{x} = \begin{bmatrix}\boldsymbol{x_1}\\\boldsymbol{x_2}\end{bmatrix}$$

where $$(\boldsymbol{x_1}|\boldsymbol{x_2}=a)\sim N(\boldsymbol{\bar \mu},\boldsymbol{\bar \Sigma})$$

How would I find the weighted conditional mean? For example, if $\boldsymbol{x_2} = \begin{bmatrix}1&2\\3&4\end{bmatrix}$, is there a way to weight the $\begin{bmatrix}1&2\end{bmatrix}$ vector to have more weight than the $\begin{bmatrix}3&4\end{bmatrix}$ vector?

• You need to define the question more precisely - how do you weight them and what is the "x-matrix", Now it is too vague for people to follow. – BGM Sep 13 '16 at 5:23
• I have added more content. – Paul Terwilliger Sep 13 '16 at 18:30
• both $\mathbf{x}_1, \mathbf{x}_2$ are random vectors following different multivariate normal distribution? Sorry not quite following your example - why suddenly equate it with a deterministic matrix, and not sure what "weight" you mean. – BGM Sep 14 '16 at 12:35