Titel_SS06

Moments of Random Variables and the Expectation Operator

Probability distributions may be defined in terms of their parameters or moments. Often
cumulative distribution functions and probability density functions are written as
and

FxX(, )p

fX(, )xp respectively to indicate the parameters p (or moments) defining the functions.

The ith moment mi of a continuous random variable is defined by:

mxfxdiXi ()

   

 x

j

(2.24)

and for a discrete random variable by:

1

()

n i

ijX

j

mxpx



 (2.25)

The mean (or expected value) of continuous and discrete random variables, X, are defined
accordingly as the first moment, i.e.:

X EX xf xdx



   

 X

Xj

(2.26)



1

()

n

Xj

j

 EX x p x



 (2.27)

where E denotes the expectation operator.

Similarly the variance, X^2 , is described by the second central moment, i.e. for continuous
random variables it is:




222
XXXVar X E ( - X ) x- f x dx


 X

(^2) )
Xj

(2.28)

and for discrete random variables as:

(^2) 
1

()(

n

XjX

j

Var X x p x


 (2.29)

where Var X  denotes the variance of X.

The ratio between the standard deviation X and the expected value X of a random variable
X is denoted the coefficient of variation CoV X  and is given by:

 X

X

CoV X 



 (2.30)

The coefficient of variation provides a useful descriptor of the variability of a random variable
around its expected value.

Probability Density and Distribution Functions

In Table 2.7 a selection of probability density and cumulative distribution functions is given
with the definition of their distribution parameters and moments.