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4-12 LOGNORMAL DISTRIBUTION 137

What lifetime is exceeded by 99% of lasers? The question is to determine x such that
P1X x2 0.99 . Therefore,

ln 1x2 10
P1X x2 P3exp1W 2 x4 P3W ln 1x24 1 a b 0.99
1.5

From Appendix Table II, 1 1z2 0.99 when z 2.33 . Therefore,

ln 1x2 10
2.33 and x exp16.5052 668.48 hours.
1.5

Determine the mean and standard deviation of lifetime. Now,
2
E1X 2 e
2 exp 110
1.1252 67,846.3
V1X2 e 2
2 1e 2 12 exp 120
2.2523exp 12.252 14 39,070,059,886.6

so the standard deviation of X is 197,661.5 hours. Notice that the standard deviation of life-
time is large relative to the mean.

EXERCISES FOR SECTION 4-12

4-117. Suppose that X has a lognormal distribution with Determine the parameters and 2 of the lognormal distribu-
2
2
parameters 5 and 9 . Determine the following: tion. (Hint: define x exp1 2 and y exp1 2 and write two
(a) P1X 13,3002 equations in terms of x and y.)
(b) The value for x such that P1X x2 0.95 4-122. The lifetime of a semiconductor laser has a log-
(c) The mean and variance of X normal distribution, and it is known that the mean and stan-
4-118. Suppose that X has a lognormal distribution with dard deviation of lifetime are 10,000 and 20,000, respec-
2
parameters 2 and 9 . Determine the following: tively.
(a) P1500 X 10002 (a) Calculate the parameters of the lognormal distribution
(b) The value for x such that P1X x2 0.1 (b) Determine the probability that a lifetime exceeds 10,000
(c) The mean and variance of X hours
(c) Determine the lifetime that is exceeded by 90% of lasers
4-119. Suppose that X has a lognormal distribution with pa-
2
rameters 2 and 4 . Determine the following: 4-123. Derive the probability density function of a lognor-
mal random variable from the derivative of the cumulative
(a) P1X 5002
distribution function.
(b) The conditional probability that X 1500 given that
X 1000
(c) What does the difference between the probabilities in
parts (a) and (b) imply about lifetimes of lognormal ran- Supplemental Exercises
dom variables?
4-124. Suppose that f 1x2 0.5x 1 for 2 x 4.
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- Determine the following:
2
mal random variable with parameters 0.5 and 1 . (a) P1X 2.52
(a) What is the probability that a page is viewed for more than (b) P1X 32
10 seconds? (c) P12.5 X 3.52
(b) What is the length of time that 50% of users view the page? 4-125. Continuation of Exercise 4-124. Determine the
(c) What is the mean and standard deviation of the time until cumulative distribution function of the random variable.
a user moves from the page? 4-126. Continuation of Exercise 4-124. Determine the
4-121. Suppose that X has a lognormal distribution and that mean and variance of the random variable.
the mean and variance of X are 100 and 85,000, respectively.