Sometimes expectations and random variables appear in so many ways that I find it a little confusing at times. The book I am following has a neat diagram that helps quite a bit.

## Basic Expectation

Start with a distribution for sample space and a random variable . The basic expectation tells us the value is likely to take *on average*.

Note that induces a decomposition of as follows

## Conditional Expectation

Instead of we could be given a distribution with respect to an event . The expectation over this is the value is likely to take *on average* conditioned on the event (i.e. restricted to).

Let’s suppose that we have an event and its conditional probabilities where is a decomposition of . We can write this as a random variable that takes on the value whenever .

Now that we have this random variable, we can once again take its expectation which in this case gives the probability of *on average* conditioned on an event from . This has a special name and is called the *total probability*.

## One More Generalization

Suppose now that we have a random variable inducing the decomposition where . We also have conditional probabilities . We can certainly take the following expectation from before

which is the expectation of conditioned on . We now do this for the entire decomposition to arrive at this random variable

which takes on a conditional expectation of at each . We can now generalize the total probability formula such that it gives the the expectation of *on average* conditioned on an event from .

The lesson here is to always translate the things we do with probabilities to expectations.