Theory

Theory 1

Expectation conditioned by a fixed event

Suppose is a random variable and . The expectation of conditioned on describes the typical value of given the hypothesis that is known.

Discrete case:

Continuous case:

Conditional variance:

Division into Cases / Total Probability applied to expectation:

Linearity of conditional expectation:

Extra - Proof: Division of Expectation into Cases

We prove the discrete case only.

1. Expectation formula:

2. Division into Cases for the PMF:

3. Substitute in the formula for :

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Expectation conditioned by a variable event

Suppose and are any two random variables. The expectation of conditioned on describes the typical of value of in terms of , given the hypothesis that is known.

Discrete case:

Continuous case:

Theory 2

Expectation conditioned by a random variable

Suppose and are any two random variables. The expectation of conditioned on is a random variable giving the typical value of on the assumption that has value determined by an outcome of the experiment.

In other words, start by defining a function :

Now is defined as the composite random variable .

Considered as a random variable, takes an outcome , computes , sets , then returns the expectation of conditioned on .

Notice that is not evaluated at , only is.

Because the value of depends only on , and not on any additional information about , it is common to represent a conditional expectation using only the function .

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Iterated Expectation

Proof of Iterated Expectation, discrete case