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Expectation

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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 X has pdf f X x (correctly normalised so that x ∞ ∞ f X x 1 ), its expectation is given by:

X x ∞ ∞ x f X x X
For discrete processes, we substitute this previous equation in here to get
X x ∞ ∞ x i 1 M p X x i δ x x i i 1 M x i p X x i X
Now, what is the mean value of some function, Y g X ?

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

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

Note, expectation is a Linear Operator :

a g 1 X b g 2 X a g 1 X b g 2 X

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Source:  OpenStax, Random processes. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10204/1.3
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