Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 308

The corresponding distribution function is:

ð

f X ðxÞ d x ¼ l ð

x x

e lx d x: F X ðxÞ¼PfX xg¼ (5.61)

0 0

From ( 5.61 ), we have:

1 e lx

x 0 ;

for

F X ðxÞ¼ (5.62)

0 for x< 0 :

Exercise E.2.13 shows that the exponential random variable X can be obtained by
the following transformation of the uniform random variable Y in the interval [0, 1]:

1
l ln ð 1 YÞ:

X ¼ (5.63)

An exponential variable with the parameter l ¼ 0.1, which is generated using
( 5.63 ), is shown in Fig. 5.8 . The corresponding density and distribution functions
are shown in Fig. 5.9a , b. Note that the low values of the random variable are much
more probable than the high values.
Exponential random variables are used in the study of queuing systems to
describe the time between arrivals of customers [MIL04, p. 64]. Additionally, the
exponential density is useful in describing the fluctuations in signal strength
received by radar from aircrafts [PEE93, p. 56].

Exponential random variable

80

70

60

50

40

30

20

10

0 0

100 200 300 400 500

n

600 700 800 900 1000

Fig. 5.8 Exponential variable with the parameter l ¼ 0.1