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:


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


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. ...

Get Discrete Stochastic Processes and Optimal Filtering now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.