Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 350

PROBABILITY: P2=P(B1<x<B2)

0.4

0.3

0.2

0.1

0 0

5 10 15 20 25

Range

PROBABILITY: P2, P2 = F(B2)-F(B1)

1

F2

0.5

F1

0

0 5 10 15 20 25

Range

Fig. 5.23 Probability P 2

Second case : The values of the normal standard deviations are decreased: s 1 ¼ 0.2,
s 2 ¼ 0.1, s 3 ¼ 0.08, s 4 ¼ 0.05. Figure 5.25a shows the values of the variables,
while Fig. 5.25b shows their corresponding estimated densities.

Exercise M.5.4 (MATLAB file: exercise_M_5_4.m ). Show that the sum of the
independent lognormal variables is a lognormal variable.
The parameters of variables are s 1 ¼ 0.6325, s 2 ¼ 0.4472, m 1 ¼ m 2 ¼ 1.
Solution The lognormal variables and their sums are shown in Fig. 5.26a , while
their corresponding estimated densities are given in Fig. 5.26b , thus confirming that
the sum is once again the lognormal variable.

Exercise M.5.5 (MATLAB file: exercise_M_5_5.m ). Generate N ¼ 1,000
samples of a Rayleigh variable X with a parameter of s 2

¼ 1. Estimate and plot

the PDF and distribution.
Solution The plot of the Rayleigh variable is shown in Fig. 5.27 . The estimated
Rayleigh PDF and the mathematical PDF are shown in Fig. 5.28 .
Figure 5.29 shows the estimated and mathematical distributions.

Exercise M.5.6 (MATLAB file: exercise_M_5_6.m ). Find the probability that the
Rayleigh random variable with the parameter s 2 ¼ 1, is less than A , and the

probability that the variable it is in the interval [ B 1 , B 2 ], where A ¼ 1 and
B 1 ¼ 0.5, B 2 ¼ 1.5.