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 X ∈ L2 (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 .
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, 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, …, Xn are pairwise independent r.v.
Var (X1 +…+ Xn) = Var X1 +… + Var Xn
The Var Xj (always positive) and the Cov (Xj, XK) (positive or negative) can take extremely high algebraic values. ...