Applied Statistics and Probability for Engineers

4-12 LOGNORMAL DISTRIBUTION 135

4-109. Suppose that X has a Weibull distribution with

and hours. Determine the mean and vari-

ance of X.

4-110. Suppose that Xhas a Weibull distribution

and hours. Determine the following:

(a) (b)

4-111. Assume that the life of a roller bearing follows a

Weibull distribution with parameters and

hours.

(a) Determine the probability that a bearing lasts at least 8000

hours.

(b) Determine the mean time until failure of a bearing.

(c) If 10 bearings are in use and failures occur independently,

what is the probability that all 10 bearings last at least

8000 hours?

4-112. The life (in hours) of a computer processing unit

(CPU) is modeled by a Weibull distribution with parameters

and hours.

(a) Determine the mean life of the CPU.

(b) Determine the variance of the life of the CPU.

(c) What is the probability that the CPU fails before 500

hours?

 3  900

 2 10,000

P 1 X10,000 2 P 1 X 50002

 100

0.2

0.2  100

4-113.Assume the life of a packaged magnetic disk exposed

to corrosive gases has a Weibull distribution with and

the mean life is 600 hours.

(a) Determine the probability that a packaged disk lasts at

least 500 hours.

(b) Determine the probability that a packaged disk fails be-

fore 400 hours.

4-114. The life of a recirculating pump follows a Weibull

distribution with parameters , and hours.

(a) Determine the mean life of a pump.

(b) Determine the variance of the life of a pump.

(c) What is the probability that a pump will last longer than its

mean?

4-115. The life (in hours) of a magnetic resonance imagin-

ing machine (MRI) is modeled by a Weibull distribution with

parameters and hours.

(a) Determine the mean life of the MRI.

(b) Determine the variance of the life of the MRI.

(c) What is the probability that the MRI fails before 250 hours?

4-116. If Xis a Weibull random variable with  1, and

1000, what is another name for the distribution of Xand

what is the mean of X?

 2  500

 2  700

0.5

4-12 LOGNORMAL DISTRIBUTION

Variables in a system sometimes follow an exponential relationship as. If the

exponent is a random variable, say is a random variable and the distribu-

tion of Xis of interest. An important special case occurs when Whas a normal distribution.

In that case, the distribution of Xis called alognormal distribution.The name follows

from the transformation ln. That is, the natural logarithm of Xis normally dis-

tributed.

Probabilities for Xare obtained from the transformation to W, but we need to recognize

that the range of Xis. Suppose that Wis normally distributed with mean and variance

; then the cumulative distribution function for Xis

for ,where Zis a standard normal random variable. Therefore, Appendix Table II can be

used to determine the probability. Also,

The probability density function of Xcan be obtained from the derivative of F(x).

This derivative is applied to the last term in the expression for F(x), the integral of the stan-

dard normal density function. Furthermore, from the probability density function, the

mean and variance of Xcan be derived. The details are omitted, but a summary of results

follows.

F 1 x 2 0, for x0.

x 0

P cZ

ln 1 x 2 

 d^ c

ln 1 x 2 

 d

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

^2

1 0,  2 

1 X 2 W

W, Xexp 1 W 2

xexp 1 w 2

EXERCISES FOR SECTION 4-11

c 04 .qxd 5/10/02 5:20 PM Page 135 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files: