Fundamentals of Probability and Statistics for Engineers

Now, writing the pdf of Y can be written in the form

The mean and standard deviation of Y can be found either through dir ect
integration by using fY (y) or by using the relationship given by Equation (7.43)
together with fX (x). In terms of Y and ln Y , they take the forms

7.3.1 Probability Tabulations

Because of the close ties that exist between the normal distribution and the
lognormal distribution through Equation (7.43), probability calculations
involving a lognormal distributed random variable can be carried out with
the aid of probability tablesprovided for normal random variables as shown
below.
Consider the probability distribution function of Y. We have

Now, since the mean of X is lnY and its variance is^2 ln Y ,wehave:

Since FU (u) is tabulated, Equation (7.50) can be used for probability calcula-
tions associated with Y with the aid of the normal probability table.

Ex ample 7. 5. Problem: the annual maximum runoff Y of a certain river can
be modeled by a lognormal distribution. Suppose that the observed mean and
standard deviation of Y are mY 300 cfs and Y 200cfs. Determine the
probability P(Y > 400 cfs).
Answer: using Equations (7.48), parameters Y and ln Y are solutions of the
equations

Some Important Continuous Distributions 211

XˆlnY,

fY…y†ˆ

1

ylnY… 2 †^1 =^2

exp

1

2 ^2 lnY

ln^2

y

Y



; fory 0 ;

0 ; elsewhere:

8

><

>:

… 7 : 47 †



mYˆYexp

^2 lnY

2



;

^2 Yˆm^2 Y‰exp…^2 lnY† 1 Š:

9

>>=

>>

;

… 7 : 48 †

FY…y†ˆP…Yy†ˆP…Xlny†ˆFX…lny†; y 0 : … 7 : 49 †

 

FY…y†ˆFU

lnylnY

lnY



ˆFU

1

lnY

ln

y

Y



; y 0 : … 7 : 50 †

ˆ  ˆ