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Discrete Stochastic Processes and Optimal Filtering by Roger Ceschi, Jean-Claude Bertein

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1.4. Second order random variables and vectors

Let us begin by recalling the definitions and usual properties relative to 2nd order random variables.

DEFINITIONS.– Given XL2 (dP) of probability density fX, E X2 and E X have a value. We call variance of X the expression:

Var X = E X2 −(E X)2 = E(X − E X)2

We call standard deviation of X the expression images.

Now let two r.v. be X and Y ∈ L2 (dP). By using the scalar product <, > on L2(dP) defined in 1.2 we have:

images

and, if the vector Z = (X, Y)admits the density fZ, then:

images

We have already established, by applying Schwarz's inequality, that EXY actually has a value.

DEFINITION.– Given that two r.v. are X, YL2 (dP), we call the covariance of X and Y :

The expression Cov (X, Y) = EXY − EX EY.

Some observations or easily verifiable properties:

Cov (X, X) = Var X

Cov (X, Y)= Cov (Y, X)

– if λ is a real constant Var (λX) = λ2 Var X;

– if X and Y are two independent r.v., then Cov(X, Y) = 0 but the reciprocal is not true;

– if X1, …, Xn are pairwise independent r.v.

Var (X1 +…+ Xn) = Var X1 +… + Var Xn

Correlation coefficients

The Var Xj (always positive) and the Cov (Xj, XK) (positive or negative) can take extremely high algebraic values. ...

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