# For which joint distributions is a conditional expectation an additive

1. Jun 5, 2014

### estebanox

I know that, for a random vector (X,Y,Z) jointly normally distributed, the conditional expectation E[X|Y=y,Z=z] is an additive function of y and z

For what other distributions is this true?

2. Jun 5, 2014

The class of elliptical distributions (see Multivariate Statistical Theory'' by Robb Muirhead for an introductory discussion) has the property I believe you want. Specifically, if
$$\mathbf{X} = \begin{bmatrix} \mathbf{X_1} \\ \mathbf{X_2} \end{bmatrix},% \mathbf{\mu} = \begin{pmatrix} \mathbf{\mu_1} \\ \mathbf{\mu_2} \end{pmatrix},% V = \begin{bmatrix} V_{11} & V_{12} \\ V_{21} & V_{22} \end{bmatrix}$$

then

$$E[\mathbf{X_1} \mid \mathbf{X_2}] = \mathbf{\mu_1} + V_{12} V_{22}^{-1} \left(% \mathbf{X_2} - \mathbf{\mu_2}\right)$$

- the same type of relationship demonstrated by multivariate normal distributions.

Last edited: Jun 5, 2014
3. Jun 5, 2014

### estebanox

Yes. That's exactly what I need. Thanks a lot!