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 ? L ² ( dP) of probability density f _X, EX ² and EX have a value. We call variance of X the expression:

We call standard deviation of X the expression .

Now let two r.v. be X and Y ? L ² ( dP). By using the scalar product <,> on L ² ( 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 ? L ² ( dP), we call the covariance of X and Y:

The expression Cov( X, Y) = EXY ? EXEY.

Some observations or easily verifiable properties:

• if ? is a real constant Var( ? X) = ? ² Var X;

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

• if X ₁, , X _n are pairwise independent r.v.