conditional expectation

conditional expectation

Let

(

Ω

,

ℱ

,

P

)

be a

probability space

and

X

:

Ω

→

ℝ

a real

random variable

with

E

⁢

[

|

X

|

]

<

∞

.

Conditional Expectation Given an Event

Given an event

B

∈

ℱ

such that

P

⁢

(

B

)

>

0

, then we define the

conditional expectation

of

X

given

B

, denoted by

E

[

X

|

B

]

to be

E

[

X

|

B

]

:=

1

P

⁢

(

B

)

∫

B

X

d

P

.

When

P

⁢

(

B

)

=

0

,

E

[

X

|

B

]

is sometimes defaulted to

0

.

If

X

is discrete, then we can write

X

=

∑

i

=

1

∞

w

i

⁢

1

B

i

, where

1

B

i

are the

indicator functions

,

B

i

=

X

-

1

⁢

(

{

w

i

}

)

and

w

i

∈

ℝ

, then conditional expectation of

X

given

B

becomes

E

[

X

|

B

]

=

1

P

⁢

(

B

)

⁢

∫

B

(

∑

i

=

1

∞

w

i

⁢

1

B

i

)

⁢

𝑑

P

=

1

P

⁢

(

B

)

⁢

(

∑

i

=

1

∞

w

i

⁢

∫

B

1

B

i

⁢

𝑑

P

)

=

1

P

⁢

(

B

)

(

∑

i

=

1

∞

w

i

P

(

B

i

∩

B

)

)

=

∑

i

=

1

∞

w

i

P

(

B

i

|

B

)

,

where

P

(

B

i

|

B

)

is the

conditional probability

of

B

i

given

B

.

Conditional Expectation Given a Sigma Algebra

If

𝒟

⊂

ℱ

is a sub

σ

-algebra, then the

conditional expectation of

X

given

D

, denoted by

E

[

X

|

𝒟

]

is defined as follows

:

Definition

E

[

X

|

𝒟

]

is the

function

from

Ω

to

ℝ

satisfying

:

1.

E

[

X

|

𝒟

]

is

𝒟

-

measurable

2.

∫

A

E

[

X

|

𝒟

]

d

P

=

∫

A

X

d

P

, for  all

A

∈

𝒟

.

It can be shown, via

Radon-Nikodym Theorem

, that

E

[

X

|

𝒟

]

always exists and is unique almost everywhere: any two

𝒟

-measurable random variables

Y

,

Z

with

∫

A

Y

⁢

𝑑

P

=

∫

A

Z

⁢

𝑑

P

=

∫

A

X

⁢

𝑑

P

differ by a null event in

𝒟

. We can in fact set up an

equivalence relation

on the set of all

integrable

𝒟

-

measurable functions

satisfying condition 2 above. In this sense,

E

[

X

|

𝒟

]

is an

equivalence class

of random variables, and any two members in

E

[

X

|

𝒟

]

may qualify as conditional expectations of

X

given

𝒟

(they are often called

versions

of the conditional expectation). In practice, however, we often think of

E

[

X

|

𝒟

]

as a function rather than a set of functions. As long as we realize that any two such functions are equal almost surely, we may blur such

differences

and abuse the

language

.

Suppose

Y

:

Ω

→

ℝ

is another random variable with

E

⁢

[

|

Y

|

]

<

∞

and let

α

,

β

∈

ℝ

. Then

1.

E

[

α

X

+

β

Y

|

𝒟

]

=

α

E

[

X

|

𝒟

]

+

β

E

[

X

|

𝒟

]

2.

E

[

E

[

X

|

𝒟

]

]

=

E

[

X

]

3.

E

[

X

|

𝒟

]

=

X

if

X

is

𝒟

-measurable

4.

E

[

X

|

𝒟

]

=

E

[

X

]

if

X

is

independent

(

http://planetmath.org/IndependentSigmaAlgebras

) of

𝒟

5.

E

[

Y

X

|

𝒟

]

=

Y

E

[

X

|

𝒟

]

if

Y

is

𝒟

-measurable

Conditional Expectation Given a Random Variable

Given any real random variable

Y

:

Ω

→

ℝ

, we define the

conditional expectation of

X

given

Y

to be the conditional expectation of

X

given

ℱ

Y

, the

sigma algebra

generated by

Y

(

http://planetmath.org/MathcalFMeasurableFunction

).

Title

conditional expectation

Canonical name

ConditionalExpectation

Date of creation

2013-03-22 15:43:45

Last modified on

2013-03-22 15:43:45

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

13

Author

georgiosl (7242)

Entry type

Definition

Classification

msc 60-00

Classification

msc 60A10

Related topic

ConditionalProbability

Related topic

ConditionalExpectationUnderChangeOfMeasure

Related topic

ConditionalExpectationsAreUniformlyIntegrable

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