Multivariate Probability Distribution with Linear Conditional Expectation I want to know what probability distribution has the linearity property of the conditional expectation.
To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$.
Then, if $[v_1,\;v_2,\;v_3]$ follows a joint normal distribution, we can show that $\mathbb{E}[v_1|v_2,\;v_3]$ is linear in $v_2$ and $v_3$.
That is, $\mathbb{E}[v_1|v_2,\;v_3]=\rho'[v_2,\;v_3]'$, where $\rho$ is a $2\times1$ constant vector consists of covariance of $[v_1,\;v_2,\;v_3]$.
Here, according to a textbook, there are diverse probability distributions that have the same linearity of the conditional expectation.
But, I am not sure what distributions also have that property.
Could you give me some examples (with proof if possible).

Suppose that jointly continuous random variables $X$ and $Y$ have joint pdf with constant value $2$ on the triangle with vertices $(0,0), (1,1), (0,1)$ in the plane. Note that this is the joint pdf of $X = \min(U,V)$ and $Y=\max(U,V))$ where $U$ and $V$ are i.i.d, $\mathcal U(0,1)$ random variables. Then, $E[X\mid Y] = \frac 12 Y$ and $E[Y\mid X] = \frac 12 + \frac 12 X$ both are linear functions of $Y$ and $X$ respectively.
As to whether the joint distribution of the minimum and maximum of two independent $\mathcal U(0,1)$ random variables is "a typical well-known distribution" or not is something that I leave for the cognoscenti on this group to decide. I am just an engineer and a mere dabbler in statistics, not a statistician
