November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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5.13 PROPERTIES OF CONDITIONAL EXPECTATION
Next, we present several properties of conditional expectation, many of which are straightforward extensions of those for ordinary expectation.
- Linearity. For constants
:
(5.126) 
This result is obvious from the definition of expectation. - Independence. For independent X and Y,
(5.127)
Since fY|X(y|x) = fY(y) for every x, the conditional expectation simplifies to
(5.128) 
- Function of conditioning variable. For g(Y) a function of Y, we have the following three results:
(5.129)
(5.130) 
(5.131) 
The first expression is intuitively obvious, but we derive it starting with the following conditional cdf:
(5.132) 
where u(y) is the unit-step function. This ...
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