Covariance
The covariance of random variables and is
a measure of the dependency between them.
Definition 3.23 (Covariance)
Let and be random variables. The covariance of and is defined by
Properties
Proposition 3.24 (Properties Of Covariance)
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Let . Then and .
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Let . Then .
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Let be random variables. Then .
More generally, for all , we have
In particular,
Proof of the product formula and the variance of a sum
For the product formula,
For the variance of a sum,
By bilinearity, .
Independence implies zero covariance
Zero covariance does not imply independence
Variance of a sum of independent random variables
Corollary 3.29
Let be independent random variables. Then
Related
Stated in
- Definition 3.23 (Covariance)§3.2 Variance and Covariance
- Proposition 3.24 (Properties Of Covariance)§3.2 Variance and Covariance
- Lemma 3.27§3.2 Variance and Covariance
- Example 3.28§3.2 Variance and Covariance
- Corollary 3.29§3.2 Variance and Covariance