Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 249

2 =1

m=0, s

4

2

0

2

-

- 4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

n

0.4

0.3

0.2

0.1

0

- 4 - 3 - 2 - 1 0

cells

1 2 3 4

Fig. 4.12 Normal random variable and its estimated PDF

Consider that the variable X (generated using the MATLAB file randn.m )isa
normal random variable with a mean value of 0 and a variance of 1. From ( 4.94 ), it
follows:

s Y ¼ a 2

m Y ¼ b ; : (4.95)

From ( 4.93 ) and ( 4.95 ), we get a linear transformation that has to be applied to
the normal variable X in order to obtain a normal variable with the desired mean
value and variance ( m Y and s Y , respectively) of length N

Y ¼ s Y X þ m Y : (4.96)

Using ( 4.93 ) the MATLAB file r ¼ fnorm ( N,VA,ME ) generates the desired
normal variable r with a length of N and the given variance VA and mean value ME .
Example 4.4.2 Generate normal variables with the variance s 2 ¼ 4 and four

different mean values: 6, 2, 3, 8. Estimate the corresponding PDFs.

Solution The generated variables are shown in Fig. 4.13a , and the corresponding
estimated densities are shown in Fig. 4.13b . Note that the estimated densities have
the same shape and are only shifted along the x -axis. Similarly, according to the “3 s
rule” all signals have the same range around their mean values.