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Expectation

Module by: Nick Kingsbury

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 xdx=1 x f X x 1 ), its expectation is given by:
EX=∫-∞∞x f X xdx=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 δx- x i dx=∑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=gX Y g X ?
Using the result of this previous equation for pdfs of related processes YY and X X:
f Y yⅆy= f X xⅆx f Y y ⅆ y f X x ⅆ x (3)
Hence (again assuming infinite integral limits unless stated otherwise)
EgX=EY=∫y f Y ydy=∫gx f X xdx 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=aE g 1 X+bE g 2 X a g 1 X b g 2 X a g 1 X b g 2 X (5)

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