Variance
The variance of a random variable is
a measure of how concentrated the distribution of is around its expectation.
Definition 3.19 (Variance)
The variance of is defined by
Standard deviation
Definition 3.20 (Standard Deviation)
The standard deviation of is defined by .
Properties
Proposition 3.21 (Propositions Of )
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.
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If , then .
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If , then and .
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.
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, with the minimum attained at .
Proof of the computational formula and the minimum property
For the computational formula,
For the minimum property, define . Then
which vanishes exactly at , and , so is convex and the minimum is attained there:
Examples
For : , and
so
For : , and
so .
Related
Stated in
- Definition 3.19 (Variance)§3.2 Variance and Covariance
- Definition 3.20 (Standard Deviation)§3.2 Variance and Covariance
- Proposition 3.21 (Propositions Of )§3.2 Variance and Covariance