Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 295

Chapter 5
Other Important Random Variables

5.1 Lognormal Random Variable

5.1.1 Density Function

A lognormal variable X is obtained through the exponential transformation of a
normal random variable Y that has a mean value of m Y and a variance of s Y ,

X ¼ e Y ; (5.1)

or

x ¼ e y ; x> 0 ; (5.2)

where x and y are ranges of the variables X and Y , respectively.
From ( 5.2 ), we have only one solution:

y ¼ ln x: (5.3)

The relation ( 5.3 ) explains why the name “lognormal” comes for the variable X .
The PDF of the variable X from ( 5.1 ) is obtained using ( 2.140 ):

y ¼ ln x

f Y ðyÞ

f X ðxÞ¼ : (5.4)

dx

j

dy ðyÞj

The derivative of the transformation ( 5.2 ) is:

d x
d y ¼ e y

¼ x; x> 0 : (5.5)