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    By: Jan E. OdegardAs a part of collection: "Random Processes"

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Expectation

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces expectation.

Expectations form a fundamental part of random signal theory. In simple terms the Expectation Operator calculates the mean of a random quantity although the concept turns out to be much more general and useful than just this.

If XX has pdf f X ⁢x f X x (correctly normalised so that ∫−∞∞ f X ⁢xd x =1 x f X x 1 ), its expectation is given by:

EX=∫−∞∞x⁢ f X ⁢xd x =X- X x x f X x X
(1)
For discrete processes, we substitute this previous equation in here to get
EX=∫−∞∞x⁢∑ i =1M p X ⁢ x i ⁢δ⁢x− x i d x =∑ i =1M x i ⁢ p X ⁢ x i =X- X x x i 1 M p X x i δ x x i i 1 M x i p X x i X
(2)
Now, what is the mean value of some function, Y=g⁢X Y g X ?

Using the result of this previous equation for pdfs of related processes YY and X X:

f Y ⁢y⁢ⅆ⁢y= f X ⁢x⁢ⅆ⁢x f Y y ⅆ y f X x ⅆ x
(3)
Hence (again assuming infinite integral limits unless stated otherwise)
Eg⁢X=EY=∫y⁢ f Y ⁢yd y =∫g⁢x⁢ f X ⁢xd x g X Y y y f Y y x g x f X x
(4)
This is an important result which allows us to use the Expectation Operator for many purposes including the calculation of moments and other related parameters of a random process.

Note, expectation is a Linear Operator:

Ea⁢ g 1 ⁢X+b⁢ g 2 ⁢X=a⁢E g 1 ⁢X+b⁢E g 2 ⁢X a g 1 X b g 2 X a g 1 X b g 2 X
(5)

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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