Page 157 - Applied Statistics And Probability For Engineers

c04.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:

4-12 LOGNORMAL DISTRIBUTION 135

EXERCISES FOR SECTION 4-11
4-109. Suppose that X has a Weibull distribution with 4-113. Assume the life of a packaged magnetic disk exposed
0.2 and 100 hours. Determine the mean and vari- to corrosive gases has a Weibull distribution with 0.5 and
ance of X. the mean life is 600 hours.
4-110. Suppose that X has a Weibull distribution 0.2 (a) Determine the probability that a packaged disk lasts at
and 100 hours. Determine the following: least 500 hours.
(a) P1X 10,0002 (b) P1X 50002 (b) Determine the probability that a packaged disk fails be-
fore 400 hours.
4-111. Assume that the life of a roller bearing follows a
Weibull distribution with parameters 2 and 10,000 4-114. The life of a recirculating pump follows a Weibull
hours. distribution with parameters 2 , and 700 hours.
(a) Determine the probability that a bearing lasts at least 8000 (a) Determine the mean life of a pump.
hours. (b) Determine the variance of the life of a pump.
(b) Determine the mean time until failure of a bearing. (c) What is the probability that a pump will last longer than its
(c) If 10 bearings are in use and failures occur independently, mean?
what is the probability that all 10 bearings last at least 4-115. The life (in hours) of a magnetic resonance imagin-
8000 hours? ing machine (MRI) is modeled by a Weibull distribution with
4-112. The life (in hours) of a computer processing unit parameters 2 and 500 hours.
(CPU) is modeled by a Weibull distribution with parameters (a) Determine the mean life of the MRI.
3 and 900 hours. (b) Determine the variance of the life of the MRI.
(a) Determine the mean life of the CPU. (c) What is the probability that the MRI fails before 250 hours?
(b) Determine the variance of the life of the CPU. 4-116. If X is a Weibull random variable with 1, and
(c) What is the probability that the CPU fails before 500 1000, what is another name for the distribution of X and
hours? what is the mean of X?

4-12 LOGNORMAL DISTRIBUTION

Variables in a system sometimes follow an exponential relationship as x exp1w2 . If the
exponent is a random variable, say W, X exp1W2 is a random variable and the distribu-
tion of X is of interest. An important special case occurs when W has a normal distribution.
In that case, the distribution of X is called a lognormal distribution. The name follows
from the transformation ln 1X2 W . That is, the natural logarithm of X is normally dis-
tributed.
Probabilities for X are obtained from the transformation to W, but we need to recognize

that the range of X is 10, 2 . Suppose that W is normally distributed with mean and variance
2 ; then the cumulative distribution function for X is

F1x2 P3X x4 P3exp1W 2 x4 P3W ln 1x24
ln 1x2 ln 1x2
P cZ d c d

for x 0 , where Z is a standard normal random variable. Therefore, Appendix Table II can be
used to determine the probability. Also, F1x2 0, for x 0.
The probability density function of X can 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 X can be derived. The details are omitted, but a summary of results
follows.