# 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).

• There is a huge number of such distributions one can construct. See stats.stackexchange.com/questions/257779 for an example.
– whuber
Jul 10, 2022 at 16:22
• @whuber Thank you for your comment every time. Your answers make me inspired and are always informative :) Jul 11, 2022 at 4:03

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
• @M.C.Park You asked "Could you give me some examples (with proof if possible)?" and so I gave you an example. I didn't prove that $E[X\mid Y]$ and $E[Y\mid X]$ are linear functions as you want, but then, neither did the answer that you have accepted. Jul 10, 2022 at 11:42