Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 300

Comparing ( 5.23 ) and ( 5.1 ), it follows that the random variable X from ( 5.23 )is
a lognormal variable which itself represents the product of a number of independent
variables, as can be seen in ( 5.19 ).
As a result, as the sum of a large number of independent random variables
approaches a normal random variable (central limit theorem) the product of a large
number of independent random variables gives rise to a lognormal variable .
The lognormal PDF can be used to describe the life distribution of many
semiconductor components, the failure of which is a result of cracks caused by
material fatigue [KLA89, p. 51].

5.2 Rayleigh Random Variable

5.2.1 Density Function

The Rayleigh density function is defined as:

f X ðxÞ¼kx k x 2

; for x 0

2

(5.24)

f X ðxÞ¼ 0 ; for x < 0

where k is a positive constant.
The density function is also expressed using the parameter s 2 , which is related
with the constant k as follows:

s 2 ¼ 1 =k; (5.25)

resulting in:

1

s 2 x x 2

f X ðxÞ¼ ; for x 0

2 s 2

(5.26)

f X ðxÞ¼ 0 ; for x < 0

The Rayleigh random variable and the corresponding density for the parameter

s 2

¼ 4 are shown in Fig. 5.3 .
The Rayleigh density is useful in describing the envelope of a narrowband normal
noise. It is also important in the analysis of measurement errors [PEE93, p. 56].
Note that the random variable has only positive values and that the PDF is
asymmetrical and has a maximum value for

p k

x ¼ s ¼ 1 = : (5.27)

r k

e

¼ f X ðsÞ¼

1 1

s p ¼

f X p k : (5.28)