Applied Statistics and Probability for Engineers

136 CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

The parameters of a lognormal distribution are and , but care is needed to interpret that

these are the mean and variance of the normal random variable W. The mean and variance of

Xare the functions of these parameters shown in (4-25). Figure 4-28 illustrates lognormal dis-

tributions for selected values of the parameters.

The lifetime of a product that degrades over time is often modeled by a lognormal ran-

dom variable. For example, this is a common distribution for the lifetime of a semiconductor

laser. A Weibull distribution can also be used in this type of application, and with an appro-

priate choice for parameters, it can approximate a selected lognormal distribution. However,

a lognormal distribution is derived from a simple exponential function of a normal random

variable, so it is easy to understand and easy to evaluate probabilities.

EXAMPLE 4-26 The lifetime of a semiconductor laser has a lognormal distribution with hours and

hours. What is the probability the lifetime exceeds 10,000 hours?

From the cumulative distribution function for X

 a

ln 1 10,000 2  10

1.5

b 1  1 0.52 2  1 0.300.70

P 1 X10,000 2  1 P 3 exp 1 W 2 10,000 4  1 P 3 W ln 1 10,000 24

1.5

 10

 ^2

Let Whave a normal distribution mean and variance ; then is a log-

normal random variablewith probability density function

The mean and variance of Xare

E 1 X 2 e

 (4-25)

(^2 2)

and V 1 X 2 e^2 



2

1 e

2

 12

f 1 x 2 

1

x 12

exp c

1 ln x 22

2 ^2

d 0 x

 ^2 Xexp 1 W 2

• 0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0123456

f(x)

x

(^2) = 0.25
(^2) = 1
ωω^2 = 2.25
ω
Figure 4-28 Lognormal probability density functions with
 0 for selected values of ^2.
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