Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 339

Finally, from ( 5.209 ), ( 5.214 ), and ( 5.215 ), we get:

s 2

Y ¼ 4 þ 8 s 2

þ 8 s 4 ð 2 þ 2 s 2

2

¼ 4 s 4

Þ : (5.216)

Exercise 5.4 The random voltage is described using a Rayleigh density function,

5 x e x 5

2

for x 0 ;

f X ðxÞ¼ (5.217)

0 otherwise :

Find the mean power of X , in a resistance of 1

O

,

P ¼ X 2 : (5.218)

Answer The mean squared value of X is

1 1

2
5

2
5

x 2 x e x 5 d x ¼ x 3 e x 5 d x:

P ¼ X 2 ¼ (5.219)

0 0

Using integral 4 from Appendix A , we arrive at:

5 2
2 ¼ 5W

2
5

P ¼ : (5.220)

Exercise 5.5 The power reflected from an aircraft to radar is described using the
exponential random variable:

f X ðxÞ¼l e lx ; (5.221)

where l is a parameter.
Find the probability that the power received by radar is less than the mean
power P 0 .
Answer From ( 5.65 ), the mean power is

1
l :

P 0 ¼ (5.222)

The desired probability is:

=lÞ¼ 1 e 1

PfX< 1 =lg¼F X ð 1 ¼ 0 : 6321 : (5.223)