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Calculus of covariances

Collected here are some identites useful in calculations of covariances between random quantities. Definition:

$$(a,b) = {\mathfrak{M}}[(a-{\mathfrak{M}}[a])(b-{\mathfrak{M}}[b])] ={\mathfrak{M}}[a b]-{\mathfrak{M}}[a]{\mathfrak{M}}[b],$$ (109)

where the $\mathfrak{M}[\ ]$ is a mathematical expectation operator. By definition of variance ${\mathfrak{D}}[a]$,

$${\mathfrak{D}}[a] = (a,a)$$ (110)

The following identities are useful :

$$(b,a) = (a,b)$$ (111)

$$(a,\gamma b) = \gamma (a,b)$$ (112)

$$(a,c+d) = (a,c)+ (a,d)$$ (113)