Random Variable - Random Signals and Processes Primer with MATLAB - page 96

(b) The probability of event A 2 (see Fig. 2.46 ) is:

PfA 2 g¼PfX> 1 g¼ 4 = 7 : (2.313)

From ( 2.309 ), using ( 2.313 ), we arrive at:

Ð

7 Ð

5 5

xf X ðxÞ d x 1 = x d x

1 1

EfXjA 2 g¼ ¼

7 ¼ 3

: (2.314)

PfA 2 g 4 =

2.8 Moments

The mean value gives little information about a random variable; that is, it
represents a sort of central value around which a random variable takes its values.
However, it does not tell us, for example, about the variation, or spread of these
values. In this section, we introduce different parameters based on the mean value
operation, which describe the random variable more completely.

2.8.1 Moments Around the Origin

One simple type of functions whose mean value might be of interest is the power
function of the random variable X

gðXÞ¼X n for n ¼ 1 ; 2 ; 3 ; ... : (2.315)

The mean value of this function is called n th moment and it is denoted as m n :

1

m n ¼ EfX n x n f x ðxÞ d x: g¼ (2.316)

1

Similarly, the moments for discrete random variables are:

g¼ 1
i¼1

m n ¼ EfX n x i PfX ¼ x i g:

(2.317)

The zero-moment is equal to 1:

m 0 ¼ EfX 0 g¼Ef 1 g¼ 1 : (2.318)