Can anyone give me an example

Question

Can anyone give me an example of two random variables $A$ and $B$ defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ which are not independent and for which we even though have $\mathbb{E}[A\mid B]=\mathbb{E}[A]$ . Many thanks for your time.

Answer 1

I am just going to give you an outline, and leave the task of filling the details to you. The construction is inspired by this question.

Let $X$ be distributed according to your favorite non-constant non-negative distribution, say $X\sim\operatorname{Poisson}(1)$; and $Y\sim\operatorname{Rademacher}$ be a r.v. independent of $X$ ($Y$ is uniform on $\{-1,1\}$).

Then set $A\stackrel{\rm def}{=}XY$ and $B\stackrel{\rm def}{=}X$. Can you show that:

• $\mathbb{E}[A\mid B] = \mathbb{E}[A] =0$; but
• $A,B$ are not independent?

Answer 2

Construct $A$ and $B$ as so:

$B$ can take the values $0$ and $1$ each with probability $1/2$.

$A$ can take the values $-1$, $0$, and $1$. Conditioned on $B=1$, $A$ is either $-1$ or $1$ each with probability $1/2$. Conditioned on $B=0$, $A$ is $0$ with probability $1$.