Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 338

From here,

p

2ln ð 0

x
s ¼

: 5 Þ : (5.206)

From ( 5.206 ), the median is found to be:

x ¼ 1 : 774 s: (5.207)

Exercise 5.3 The random variable X is a Rayleigh variable with a parameter of s .
Find the variance of the random variable

Y ¼ 2 þ X 2 : (5.208)

Answer The variance of the random variable Y is

Y 2 s 2

Y ¼ Y 2

:

(5.209)

From ( 5.208 ), the mean and the mean squared values are:

Y ¼ 2 þ X 2 ; (5.210)

2

Y 2 ¼ð 2 þ X 2 ¼ 4 þ X 4 þ 4 X 2 Þ : (5.211)

From ( 5.35 ), the mean squared value of Rayleigh variable X is

¼ 2 s 2

X 2 :

(5.212)

The fourth moment X 4 is obtained using the integral 4 from Appendix A ,

x 2
2 s 2 d x ¼

1

1
s 2

1
s 2 8 s 6

x 5 e ¼ 8 s 4 X 4 ¼ : (5.213)

0

Placing ( 5.212 ) into ( 5.210 ), we obtain:

Y ¼ 2 þ 2 s 2 : (5.214)

Similarly,

¼ 4 þ 8 s 4 þ 8 s 2 Y 2 ¼ 4 þ X 4 þ 4 X 2 : (5.215)