Discrete Stochastic Processes and Optimal Filtering

1.4. Second order random variables and vectors

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

DEFINITIONS.– Given X ∈ L² (dP) of probability density f_X, 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 .

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 f_Z, then:

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

DEFINITION.– Given that two r.v. are X, Y ∈ L² (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, …, X_n are pairwise independent r.v.

Var (X₁ +…+ X_n) = Var X₁ +… + Var X_n

Correlation coefficients

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