Titel_SS06

 ^22222  

222 22

() 2 2

=2

XXXX

XXX

Var X E X E X X E X E X

EX EX





    

 

X

(2.37)

By application of Equation (2.37) the following properties of the variance operator Var
can easily be derived:

 

 

 

2

2

Var c 0

Var cX c Var X

Var a bX b Var X





 

(2.38)

where and ab, c are constants and X is a random variable.

From Equation (2.37) it is furthermore seen that in general it is EgX ()/gEX( ). In fact
for convex functions it can be shown that the following inequality is valid (Jensen’s
inequality):

gx()

EgX()*gEX( ) (2.39)

where the equality holds if gX() is linear.

Whether the cumulative distribution and density function are defined by their moments or by
parameters is a matter of convenience and it is generally possible to establish the one from the
other.

Random Vectors and Joint Moments

If a n-dimensional vector of continuous random variables , is considered
the joint cumulative distribution function is given by:

X 12 , ,...,

T

 XX Xn

FX

 

x 



0

PX x X x11 2 2 X xnn (2.40)

and the joint probability density function is:

 



12

X x

n

n

f

xx x



 



 X

F x (2.41)

The covariance between CXXij Xi and Xj is defined by:

CEX XXXij (- )( - )iX jXi j 

x x fxxdxdiX jX XXij i- -i

j ij

,

      

 xj (2.42)

and is also called the joint central moment between the variables Xi and Xj.

The covariance expresses the dependence between two variables. It is evident that
CVarXXXii  i. On the basis of the covariance the correlation coefficient is defined by:

Xi

ij

ij

j

XX

XX

X

C

1 (2.43)





It is seen that (^1) XXii 1. The correlation coefficients can only take values in the interval 1; 1.
A negative correlation coefficient between two random variables implies that if the outcome