Random Variable - Random Signals and Processes Primer with MATLAB - page 77

or

2 X

N

y 2 X

N

x i ¼ 2 Ny 2 X

N

x i ¼ 0 : (2.209)

i¼ 1 i¼ 1 i¼ 1

From ( 2.209 ), we arrive at:

x 1 þ x 2 þþ x N
N

y ¼ : (2.210)

Note that this is the same expression as we saw in ( 2.205 ). Therefore, the average
m av of the set of numbers x i , i ¼ 1,

, N , can be viewed as the number, which is
simultaneously closest to all the numbers in the set or the center of gravity of the set.
The next question is: “can we apply the same concept to a random variable?”

...

2.7.2 Concept of a Mean Value of a Discrete Random Variable

Consider the discrete random variable X with the values x 1 , , x N . We cannot

directly apply the formula ( 2.205 ), because in ( 2.205 ) each value x i occurs only
once, and the values x i of the random variable X occur with a certain probability. In
order to apply ( 2.205 ), we must know the exact number of occurrences for each
value x i . The required values can be obtained in one experiment.
Let the experiment be performed M times. We will measure how many times
each value x i occurs in the experiment:

...

X ¼ x 1 N 1 times,
X ¼ x 2 N 2 times,
... ...
X ¼ x N N N times

(2.211)

;

where

M ¼ N 1 þ N 2 þ ... þ N N : (2.212)

Now we have the finite set of values: N 1 values of x 1 , N 2 values of x 2 , etc.

S ¼ N 1 x 1 þ N 2 x 2 þ ... þ N N x N

(2.213)

and we can find the mean value of the set ( 2.211 ) using ( 2.205 ):

¼ X

N

S
M ¼

N 1 x 1 þþ N N x N
M

N i
M x i :

m emp ¼ (2.214)

i¼ 1