For all humankind
Academicsubsite
ZixuanZhang
ZixuanZhang
Ponder...

Conditional Expectation

For a discrete random variable and an event with ,

Given a random variable , this extends to a random variable

Definition 3.40 (Conditional Expectation)

Let be a discrete random variable and with . The conditional expectation of given and event is defined by

Definition given a random variable

Definition 3.42 (Conditional Expectation Given a Random Variable)

Let and be discrete random variables. The conditional expectation of given is defined by

where .

A random variable that is a function of Y

is a random variable, namely a function of : if , then . Random variables are functions , so really means the composition .

Tower property

Proposition 3.45 (Tower Property)

Let and be discrete random variables. Then

Proof of the tower property

Since , taking expectation gives

Independence makes the conditional expectation constant

Proposition 3.46

Proof of the independence case

If and are independent discrete random variables,

Iterated conditioning

Proposition 3.47

Proof of iterated conditioning

Write with . The claim is that is independent of :

Since and are independent, ; by the tower property . Hence

Linearity in X

Proposition 3.44

Let and be discrete random variables. Then for some constant ,

  1. .

  2. .

    In particular, .

  3. .

Pulling out known factors

Proposition 3.48

Let be two random variables and . Then

Proof of pulling out known factors

Conditioning on freezes the factor :

Idempotence

Corollary 3.49

Let be two random variables. Then

  2. .

Example: coin tossing

Toss a -coin times independently, let and . Two computations give .

Directly,

Or by symmetry: since all are equal,

so again .

Related

Stated in