Conditional Expectation

Conditional Expectation

I returned to chapter 3 this morning since I still hadn't covered conditional expectation. At first glance, the definition of conditional expectation is pretty straight forward:

$$E(X|Y=y) = \int x f_{X|Y}(x|y) dx$$

what's a bit subtle is that this is not a value, but a function of $y$. As the book states,

Whereas $E(X)$ is a number, $E(X|Y=y)$ is a function of $y$. Before we observe $Y$ we don't know the value of $E(X|Y=y)$ so it is a random variable which we denote $E(X|Y)$. In other words, $E(X|Y)$ is the random variable whose value is $E(X|Y=y)$ when $Y=y$.

I wrote up a series of examples related to conditional distributions and conditional expectation as way of both reviewing conditional expectation and solidifying the reading. It also uses The Rule of Iterated Expectations which says $E[E(Y|X)] = E(Y)$.

With this review of conditional probability in mind, I went back and looked at this problem again which considers flipping a coin $N$ times where $N \sim Poisson(\lambda)$. I previously found this problem mind blowing but it seems more straightforward thinking in terms of conditional expectation, even though the fact that the resulting random variable is also a Poisson remains pretty nifty.