Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 282

2 =4

m=0, s

10

8

6

4

2

0

- 2

- 4

- 6

- 8 0

100 200 300 400 500

n

600 700 800 900 1000

Fig. 4.35 Random variable and the interval for probability calculation

This probability is illustrated in Fig. 4.36 , as the area below the PDF in the interval
[ 1, 5]. Similarly, the probability is presented in the distribution function as the
difference of F X (5) and F X ( 1).
Exercise M.4.6 (MATLAB file: exercise_M_4_6.m ) Show that the linear trans-
formation Y ¼ 3 X + 3 of the normal random variable X ¼ N (1, 1) is also a normal
random variable.

Solution The input and output random variables along with the transformation are
shown in Fig. 4.37 .
The estimated PDFs are shown in Fig. 4.38 .
The estimated PDFs show that both random variables are normal variables. The
mean value and variance of the output variable Y are:

s 2

Y ¼ 3 2 s 2

m Y ¼ 3 m X þ 3 ¼ 6

X ¼ 9

: (4.237)

;

Exercise M.4.7 (MATLAB file: exercise_M_4_7.m ) Show that the sum of two
independent normal random variables:

X 1 ¼ Nð 1 ; 4 Þ and X 2 ¼ Nð 5 ; 8 Þ (4.238)

is also a normal random variable.