Is multivariate normal the only distribution with this property?

Question

Suppose the random scalars $X$ and $Y$ are bivariate normally distributed. Then $E[X|Y=y]$ is linear in $y$ and $E[Y|X=x]$ is linear in $x$. In other words, both conditional expectation functions are linear. What I am wondering is whether the bivariate normal distribution is the only continuous distribution with this property? If not, then is it the only distribution with this property that also has say, a continuous pdf? I know that in higher dimensions the multivariate normal has the same liner conditional expectation property, but are there other examples in that case?

Answer 1

No, the bivariate normal is not the only distribution with the property that $E[X\mid Y=y]$ is a linear function of $y$ and also that $E[Y\mid X=x]$ is a linear function of $x$; many other distributions enjoy the same property.

For example, suppose that $(X,Y)$ is uniformly distributed on the triangle with vertices $(0,0), (1,1), (0,1)$ so that the joint density $f_{X,Y}(x,y)$ has value $2$ on the interior of the triangle. As motivation, note that this is the joint pdf of $\left(\min(U,V),\max(U,V)\right)$ where $U$ and $V$ are i.i.d. $\mathcal U(0,1)$ random variables. Now, notice that given $Y=y, y \in (0,1)$, the conditional distribution of $X$ is uniform on $(0,y)$ and so $E[X\mid Y=y] = y/2$ is a linear function of $y$. Similarly, given that $X=x, x \in (0,1)$, the conditional distribution of $Y$ is uniform on $(x,1)$ and so $E[Y\mid X=x] = \frac 12 + \frac x2$ is a linear function of $x$.

Answer 2

No - it is not just a property of bivariate normals. For example

• Let $A,B,C$ be i.i.d. with finite mean $\mu$. Then let $X=A+B$ and $Y=A+C$.

• $E[A \mid X=x] =E[B \mid X=x] = \frac12 E[A+B \mid X=x]=\frac12 E[X \mid X=x]= \frac 12x$.

• So $E[Y \mid X=x]=E[A \mid X=x] +E[C \mid X=x] = \frac 12x+\mu$ which is linear in $x$.

• Similarly $E[X \mid Y=y]=E[A \mid Y=y] +E[B \mid Y=y] = \frac 12y+\mu$.