What's a real life example of a case in which the conditional expectation is not zero, but the non conditional expectation is zero?

Question

If I had to give an example to my grandmother of such a case, what would be a good real life example for a case in which

$ E[u|x]\neq 0 \quad \text{and} \quad E[u]=0 ?$

Thanks!

Answer 1

For any random variable $x$ with mean zero such that $P[x=0]\ne1$, for example $x$ standard normal, consider $u=x$. Then $E[u\mid x]=x$ almost surely hence the event $[E[u\mid x]=0]$ has probability zero, and $E[u]=E[x]$ is zero.

Add to this your favorite real-life situation modeled by a gaussian random variable.

Or, if Einstein's thought experiments count as real life, let $u$ denote the displacement of a pollen particle after $1$ minute and $x$ its displacement after $7$ minutes. Then $E[u]=E[x]=0$ and $x-u$ is independent of $u$ hence $E[x\mid u]=u+E[x-u]=u$, which is nonzero with full probability.

Or, if your grandmother is allergic to pollen, replace the pollen particle by a drunkard walking in the street, à la Pólya.

Answer 2

$E[X| X=1]$ for arbitrary $X \in \{ \pm 1\}$ with $P(X=1) =P(X=-1)=\frac{1}{2}$

This is a trivial example. Put in for $X$ getting one USD for it's raining tomorrow, and $X$ paying one USD for not.

Answer 3

Admittedly, this might be a bit of a stretch but maybe this is what you're looking for.

Because of the communications when you are to visit your grandma, it is hard for you to say exactly when you're going to be there. But a fitting description of your arrival is that it's normally distributed around 3 PM (for some reason... !). We can then view $u$ as the deviation from 3 PM that you arrive -- if you arrive at 2.50 PM $u=-10$ and so on. Its expected value is $0$, since you expect to arrive at 3 PM sharp. $x$ can then for example be the deviation from the estimated time of arrival of the bus that you're taking. Conditional on the bus, you are no longer expected to arrive at 3 PM sharp.