Applied Statistics and Probability for Engineers

4-12 LOGNORMAL DISTRIBUTION 137

What lifetime is exceeded by 99% of lasers? The question is to determine xsuch that

. Therefore,

From Appendix Table II, when. Therefore,

Determine the mean and standard deviation of lifetime. Now,

so the standard deviation of Xis 197,661.5 hours. Notice that the standard deviation of life-

time is large relative to the mean.

EXERCISES FOR SECTION 4-12

V 1 X 2 e^2 



2

1 e

2

 12 exp 120

 2.25 23 exp 1 2.25 2  14 39,070,059,886.6

E 1 X 2 e



(^2 2)
exp 110
1.125 2 67,846.3
ln 1 x 2  10
1.5

2.33 and xexp 1 6.505 2 668.48 hours.

1  1 z 2 0.99 z2.33

P 1 Xx 2 P 3 exp 1 W 2 x 4 P 3 Wln 1 x 24  1  a

ln 1 x 2  10

1.5

b0.99

P 1 Xx 2 0.99

4-117. Suppose that Xhas a lognormal distribution with

parameters and. Determine the following:

(a)

(b) The value for xsuch that

(c) The mean and variance of X

4-118. Suppose that Xhas a lognormal distribution with

parameters and. Determine the following:

(a)

(b) The value for xsuch that

(c) The mean and variance of X

4-119. Suppose that Xhas a lognormal distribution with pa-

rameters and. Determine the following:

(a)

(b) The conditional probability that given that

(c) What does the difference between the probabilities in

parts (a) and (b) imply about lifetimes of lognormal ran-

dom variables?

4-120. The length of time (in seconds) that a user views a

page on a Web site before moving to another page is a lognor-

mal random variable with parameters and.

(a) What is the probability that a page is viewed for more than

10 seconds?

(b) What is the length of time that 50% of users view the page?

(c) What is the mean and standard deviation of the time until

a user moves from the page?

4-121. Suppose that Xhas a lognormal distribution and that

the mean and variance of Xare 100 and 85,000, respectively.

0.5 ^2  1

X 1000

X 1500

P 1 X 5002

 2 ^2  4

P 1 Xx 2 0.1

P 1500 X 10002

 2 ^2  9

P 1 Xx 2 0.95

P 1 X13,300 2

 5 ^2  9

Determine the parameters and of the lognormal distribu-

tion. (Hint:define and and write two

equations in terms of xand y.)

4-122. The lifetime of a semiconductor laser has a log-

normal distribution, and it is known that the mean and stan-

dard deviation of lifetime are 10,000 and 20,000, respec-

tively.

(a) Calculate the parameters of the lognormal distribution

(b) Determine the probability that a lifetime exceeds 10,000

hours

(c) Determine the lifetime that is exceeded by 90% of lasers

4-123. Derive the probability density function of a lognor-

mal random variable from the derivative of the cumulative

distribution function.

Supplemental Exercises

4-124. Suppose that for

Determine the following:

(a)

(b)

(c)

4-125. Continuation of Exercise 4-124. Determine the

cumulative distribution function of the random variable.

4-126. Continuation of Exercise 4-124. Determine the

mean and variance of the random variable.

P 1 2.5X3.5 2

P 1 X 32

P 1 X2.5 2

f 1 x 2 0.5x 1 2 x4.

xexp 1  2 yexp 1 ^22

 ^2

c 04 .qxd 5/10/02 5:20 PM Page 137 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: