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# More on Random Variables

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Expectation  ::  Properties  ::  Pairs  ::  Independence  ::  Two R.V.S.  ::  Functions  ::  Inequalities  ::  Conditional  ::  General

## A more general definition of conditional expectation

We will explore conditional expectation in terms of probability spaces. Suppos is a sub -field of . A function is ''measurable with respect to '' if

(Any such function would be a r.v. too. But this is more restricted.)

We will now define ''conditioning on a -field.'' Suppose is a r.v. with and is a sub -field of . Then is any -measurable random variable such that

If itself were measurable it would be its own conditional expectation.

Fact: A r.v. is measurable with respect to if and only if there is a measurable function such that .

We now define conditional expectation with respect to a -field:

Properties:

1. By the fact stated above, we can write

for some function , .
2. If itself is -measurable, then .
3. .
4. If , then
5. If and , then

Idea: If you first condition on a field that is less ''course'' than you get a r.v. Then condition on .

Note: and are independent r.v.s iff and are independent -fields.

1. If is independent of then .

2. If is -measurable then .

So, for example, if , and for some (that is, is -measurable) then