Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 278

2 =12

m=0, s

0.1

0.05

0

- 20 - 15 - 10 - 5 0 5

x

10 15 20 25 30

2 =12

m=10, s

0.1

0.05

0

- 20 - 15 - 10 - 5 0 5 10 15 20 25 30

x

Fig. 4.29 PDFs of random variables with different mean values and equal variances

Exercise M.4.3 (MATLAB file: exercise_M_4_3.m ). In this exercise, we show
how different variances change the normal variables and their corresponding
densities. To this end, generate 50,000 values of two normal random variables
with equal mean values m 1 ¼ m 2 ¼ 5 and different variances s 1 ¼ 4, s 2 ¼ 64.
Solution The corresponding random variables and the PDFs are shown in
Figs. 4.30 and 4.31 , respectively. Note that the positions of the PDFs on the x -
axis are the same but have different shapes. By increasing the variance, the
maximum value of the PDF decreases and the width increases.

Exercise M.4.4 (MATLAB file: exercise_M_4_4.m ) Generate a normal random
variable X with a mean value equal to 1 and a variance equal to 4. Plot the
corresponding PDF and distribution and find the corresponding probabilities:

(a) P 1 ¼ P { X < 3}

(b) P 2 ¼ { X < 0}

Solution
(a) The first 1,000 values of the random variable along with the value of 3 are
shown in Fig. 4.32 , where the limit of 3 on the values for calculating the
probability P 1 is denoted with a dotted line. The corresponding probability is:

P 1 ¼ 0 : 84135 (4.232)