Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 337

5.9 Numerical Exercises

Exercise 5.1 Show that a Rayleigh variable has only positive values.

Answer The area below the PDF must be unity:

1

1
s 2

x e x 2

2 s 2 d x ¼ 1

; (5.199)

a

where a is the low integral limit which must be defined in order for the condition
( 5.199 ) to be satisfied.
We introduce the auxiliary variable y :

x 2
2 s 2 ¼ y:

(5.200)

From here,

x d x=s 2

¼ d y; (5.201)

Using ( 5.200 ) and ( 5.201 ), the integral ( 5.199 ) becomes:

1

a 2 = 2 s 2

1 ¼ e a 2

= 2 s 2

e y d y ¼ e y j ¼ 1 : (5.202)

a 2 = 2 s 2

From here, it follows that a ¼ 0. That is, in order to satisfy ( 5.199 ), a Rayleigh
variable must have only positive values.
Exercise 5.2 Find the mode and median for a Rayleigh variable.
Answer Mode is a x value for which the PDF has its maximum value:

0 1

x 2
2 s 2

x 2
2 s 2

¼ 0

x 2
s 2

d f X ð x Þ
d x ¼

d
d x

1 1

@

s 2 x e

A ¼

s 2 e

1 : (5.203)

From here x ¼ s , as shown in Fig. 5.3 .
Median is a value of the random variable, for which F X ( x ) ¼ 0.5.

x 2
2 s 2

5 ¼ F X ðxÞ¼ 1 e 0 : : (5.204)

From here, it follows:

x 2
2 s 2

5 ¼ e 0 : : (5.205)