**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}, *E X*^{2} and *E X* have a value. We call variance of *X* the expression:

Var *X* = E *X*^{2} −(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 ∈ *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 E*XY* 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*) = E*XY* − E*X* E*Y*.

Some observations or easily verifiable properties:

Cov (*X*, *X*) = Var *X*

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

– if *λ* is a real constant Var (λ*X*) = *λ*^{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.

Var (*X*_{1} +…+ *X*_{n}) = Var X_{1} +… + 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. ...

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