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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5.15 VARIANCE, COVARIANCE, AND CORRELATION
Measures of dispersion or spread of a random variable are considered next.
Definition: Variance The variance of random variable X is
(5.201) 
where 
is the mean.
The variance is centered in the sense that the mean of the random variable is subtracted before computing the expectation; it is one of the central moments defined in a subsequent section. The variance can be expressed in terms of the mean and the mean-square as follows:
We summarize several properties of the variance:
- Linear transformation. For aX+b:
(5.203)
where 
are constants. The variance changes when the random variable is scaled (slope a of the transformation), but is invariant to a translation (intercept b of the transformation). This is expected because a translation changes the mean and the variance is a centered expectation. - Variance of a sum. The variance of the sum of two dependent random variables X and Y is (5.204) where is the covariance CXY of X and Y (and ...
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