Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 290

Q.4.10. What is the practical usefulness of the CLT because if N is approaching
infinity, the variance is also approaching to infinity?
Q.4.11. Does the CLT also stand for a discrete random variables; knowing that
PDFs of discrete random variables have delta impulses?

4.11 Answers

A.4.1. Consider a linear transformation

Y ¼ aX þ b

(4.242)

of a random variable X , where a and b are constants.
The linear transformation ( 4.242 ) changes the mean value and the
variance of the variable X :

m Y ¼ am X þ b;
s 2

Y ¼ a 2 s 2 X :

(4.243)

Each normal variable is determined only by its two parameters: a mean
value and a variance. Therefore, the linearly transformed normal random
variable is also a normal variable with new parameters ( 4.243 ).
The uniform variable X , defined in the range [ a 1 , a 2 ], is equivalently
defined by using the length of its range

D X ¼ a 2 a 1 , and its mean value
m X - which is in the middle of the range-as shown in Fig. 4.44a .
The length of the range

D X is related to the variance as:

2
X

s X ¼ D

12 :

(4.244)

Therefore, a uniform variable, like a normal variable, is also uniquely
defined with its mean value and variance.

Fig. 4.44 Linear transformation of uniform random variable