Q:

Regression Proof

If the joint density function of $X$ and $Y$ is given by: $$f(x,y)= \begin{cases}
1/2,  & \text{for } |x| + |y| \le 1 \\ 0, & \text{otherwise}  \\ \end{cases}$$
Show that $Y$ has constant regression with respect to $X$ and/but that $X$ and $Y$ are not independant.

A:

Since $(X,Y)$ is uniform on the square with vertices $(\pm1,0)$ and $(0,\pm1)$, conditionally on $[X=x]$, $Y$ is uniform on its intersection with $\{x\}\times\mathbb R$, that is, uniform on the interval $(-1+|x|,1-|x|)$. In particular $\mathbb E(Y\mid X)=0$ while $(X,Y)$ is not independent.

