Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 296

The normal variable Y has the following PDF:

2

e ð y m Y Þ

1
2 p

2 s Y

f Y ðyÞ¼ p : (5.6)

s Y

Placing ( 5.6 ) and ( 5.5 ) into ( 5.4 ), we obtain the density function of the lognormal
variable:

2

e ð ln x m Y Þ

2 s Y

p

2 p

1

for x 0 ;

f X ðxÞ¼ (5.7)

s Y x

f X ðxÞ¼ 0 for x < 0

Note that m Y and s Y respectively, are the mean value and variance of the normal
random variable Y , and not of the lognormal variable X . Figure 5.1 shows the
lognormal variables (a) and densities (b), for different values of s Y and for
m Y ¼ 0.01. For small values of s Y ,( s Y 0.3), the lognormal PDF becomes more
symmetric, while for s Y >

0.3, the PDF becomes asymmetric.

5.1.2 Distribution Function

Using the definitions ( 2.61 ) and ( 5.7 ), the distribution function of a lognormal
variable is:

2

e ð ln x m Y Þ

ð

x

ð

x

1
2 ps Y

2 s Y

F X ðxÞ¼PfX xg¼ f X ðxÞdx ¼ p d x: (5.8)

x

1 1

However, we can express the distribution ( 5.8 ) in terms of the normal distribu-
tion and thus make possible the use of erf function to more easily derive the
lognormal distribution, as shown in the following equation:

2

e ð y m Y Þ

ð ð

ln x ln x

1
2 ps Y

2 s Y

F X ðxÞ¼PfX xg¼ f Y ðyÞdy ¼ p

d y:

(5.9)

1 1

Using the expressions for erf function ( 4.29 ) we arrive at:

8
<

ln x m Y

1
2

1 þ erf

p s Y

x 0 ;

for

F X ðxÞ¼ (5.10)

:

0 otherwise :

As an example, Fig. 5.2 shows the distribution function for m Y ¼ 0.01 and
s Y ¼ 1.