Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 299

From ( 5.14 ), taking n ¼ 1 and n ¼ 2, we obtain the mean and the mean squared
values:

m 1 ¼ m X ¼ X ¼ e m Y þ s Y =

2

; (5.15)

¼ e 2 m Y þ 2 s Y m 2 ¼ X 2 : (5.16)

The corresponding variance is obtained from ( 5.15 ) and ( 5.16 ):

X ¼ m 2 m 1 ¼ e 2 m Y þ s 2

Y ð e s 2

s 2

Y 1 Þ:

(5.17)

5.1.4 What Does a Lognormal Variable Tell Us?

Consider a large number of independent random variables

Y 1 ; ... ; Y N ; (5.18)

and the random variable X , which is equal to the product of the variables ( 5.18 )

X ¼ Y

N

Y i : (5.19)

i¼ 1

Taking the natural logarithm of both sides of ( 5.19 ), we get:

ln X ¼ X

N

ln Y i : (5.20)

i¼ 1

According to the central limit theorem, the right side of ( 5.20 ) can be

approximated as a normal random variable Y ,

Y X

N

ln Y i :

(5.21)

i¼ 1

Taking into account ( 5.21 ), the expression ( 5.20 ) becomes:

ln X Y: (5.22)

This expression can be rewritten as:

X e Y

: (5.23)