Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 242

Table 4.2 The “3 rule”

s

k Pfm ks X m þ ksg

0.5 0.3829

1 0.6827

1.5 0.8664

2 0.9545

2.5 0.9876

3 0.9973

3.5 0.9995

4 0.9999

Table 4.2 presents the values of probabilities ( 4.68 ) for different values of k.
From Table 4.2 , we note that for k ¼ 3 we have:

Pfm 3 s X m þ 3 sg¼ 0 : 9973 ; (4.69)

or, that 99.73% of all values of the normal variable are in symmetrical intervals of 3
standard deviations around the mean value. This is known as the “3 s rule.” This
rule states that the absolute value of the dissipation of a normal random variable
from its mean value effectively does not pass 3 standard deviations.

4.3 Transformation of Normal Random Variable

All transformations discussed in Sect. 2.6 can easily be applied to the normal
variable.

4.3.1 Monotone Transformation

Let us first consider the linear transformation.

4.3.1.1 Linear Transformation

Let X be a normal random variable with mean value and variance m X and s X ,
respectively, and

Y ¼ aX þ b;

(4.70)

where a and b are constants.