Random Variable - Random Signals and Processes Primer with MATLAB - page 84

Example 2.7.6 Consider the random variable X from Example 2.7.5, where we
found

EfX 2

EfXg ¼ 0 g ¼ 5 = 2 : (2.248)

;

The transformation of the variable X was a quadratic function (i.e., a nonlinear
function)

gðXÞ ¼ X 2

: (2.249)

The corresponding mean value is:

g ¼ EfX 2

E gðXÞ f g ¼ 5 = 2 : (2.250)

However, the quadratic transformation of the mean value is:

2

¼ 0 2

g EfXg ð Þ ¼ EfXg ð Þ ¼ 0 : (2.251)

Therefore,

EfgðXÞg 6¼ gðEfXgÞ: (2.252)

2.7.3 Mean Value of a Continuous Random Variable

We can apply a similar approach to that used to derive the mean value of a discrete
random variable, in order to develop the expression for the mean value of a
continuous random variable. Consider the continuous random variable X with the
continuous range x from 0 to A (Fig. 2.40 ). We divide the range x into k small
intervals Dx such that, if the random variable belongs to the i th interval [( i 1) Dx ,
iDx ], we can consider that it is equal to the end of the interval (i.e., equal to iDx ).
In other words, if Xe½ði 1 ÞDx; iDx it means that X ¼ i Dx , , k .

Imagine that we perform an experiment N times and that we obtain the number
of times that the variable X was in each of the intervals iDx , i ¼ 1,

i ¼ 1,

...

, k :

...

X ¼ Dx N 1 times,

X ¼ 2 Dx N 2 times,

... ...

X ¼ i Dx N i times,

... ...

X ¼ k Dx N k times : (2.253)