Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 359

A=0.01 A=0.5

1 1

0.5 0.5

0 0 0

5 10 0 5 10

Range Range

A=1 A=4

1 1

0.5 0.5

0 0

0 5 10 0 5 10

Range Range

Fig. 5.36 Rician distributions

Generate the corresponding variables and estimate the corresponding densities
and distributions. Compare the results of cases (a) and (b) with Rayleigh and normal
random variables, respectively. Plot the corresponding mathematical densities and
distributions.
Solution Figures 5.37 - 5.39 show results for the chosen value s 2 ¼ 4, A 1 ¼ 0.4,

A 2 ¼ 10, A 3 ¼ 2.
All cases including, A ¼ 0, are compared in Figs. 5.40 - 5.44 .

Exercise M.5.9 (MATLAB file: exercise_M_5_9.m ). Generate N ¼ 1,000
samples of an exponential variable X with parameter l ¼ 0.8. Estimate and plot
PDF and distribution.

Solution The exponential variable is shown in Fig. 5.45a . The estimated and
mathematical PDFs, along with the corresponding distribution functions are
shown in Fig. 5.45b .
Exercise M.5.10 (MATLAB file: exercise_M_5_10.m ). For the exponential vari-
able from Exercise M.5.9 find the probability that the random variable is less than
A ¼ 1, and the probability that the random variable is in the interval [0.5, 2]. Show
both probabilities in the PDF and distribution plots.

Solution Figure 5.46 illustrates the probability P 1 .

P 1 ¼ PfX 1 g¼ 0 : 5507 :