Computational formula for the variance
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In probability theory, the computational formula for the variance Var(X) of a random variable X is the formula
where E(X) is the expected value of X. This formula can be generalized for covariance:
as well as for the n by n covariance matrix of a random vector of length n:
and for the the n by m cross-covariance matrix between two random vectors of length n and m:
where expectations are taken element-wise and and are random vectors of respective lengths n and m (not necessarily equal).
[edit] Proof
The computational formula for the variance follows in a straightforward manner from the linearity of expected values and the definition:
This is often used to calculate the variance in practice.
[edit] Applications
Its applications in systolic geometry include Loewner's torus inequality.