PROBABILITY 12

Conditional Expectation

Conditional expectation is the expected value calculated using probabilities updated by known information.

The basic idea

Ordinary expectation averages over all possible outcomes. Conditional expectation averages after restricting attention to information that is known.

\[\boxed{\text{information}\rightarrow\text{conditional distribution}\rightarrow\text{conditional mean}}\]

1. Conditioning on an event

For a discrete random variable \(X\) and event \(A\) with \(P(A)>0\),

\[\boxed{E[X\mid A]=\sum_x xP(X=x\mid A)}.\]

2. Conditioning on another random variable

We can also condition on \(Y=y\):

\[E[X\mid Y=y].\]

This gives the mean of \(X\) under the conditional distribution determined by the observed value of \(Y\).

3. Conditional expectation as a random variable

The notation

\[E[X\mid Y]\]

represents a function of the random variable \(Y\). Before \(Y\) is observed, this conditional expectation can itself vary.

4. Law of total expectation

A fundamental identity is

\[\boxed{E[E[X\mid Y]]=E[X]}\]

We can first average within conditional groups and then average those conditional means to recover the overall expectation.

5. Example with events

If events \(A_1,\ldots,A_n\) partition the sample space, then

\[E[X]=\sum_i E[X\mid A_i]P(A_i).\]

6. Why it matters for stochastic processes

Conditional expectation formalises prediction using currently available information. It is fundamental to Markov processes, martingales and stochastic calculus.

7. Connection to the next chapter

A stochastic process is a family of random variables indexed by time. Conditional probabilities and expectations allow us to describe how information about the present affects predictions of the future.

\[\boxed{\text{probability foundations}\rightarrow\text{stochastic processes}}\]