Page 169 - Applied statistics and probability for engineers

Section 4-11/Lognormal Distribution 147

Example 4-26 Semiconductor Laser The lifetime (in hours) of a semiconductor laser has a lognormal distribu-
tion with θ = 10 and ω = .5. What is the probability that the lifetime exceeds 10,000 hours?
1
From the cumulative distribution function for X,
(
1
P X >10 ,000) = − P ⎡exp W ( ) ≤ 10 ,000⎤ ⎦
⎣
⎡
= 1 − P W ≤ ( 10 ,000)⎤ ⎦
ln
⎣
(
⎛ ln 110 000) − 10⎞
,
0 52)
(
= 1 − Φ ⎜ ⎝ 1 5 ⎟ ⎠ = 1− Φ − .
.
= − . 0 70
1 0 30 = .
(
What lifetime is exceeded by 99% of lasers? The question is to determine x such that P X > x) = .0 99 . Therefore,
(
x)
⎡
P X > x) = P ⎡exp ( ) ⎦ P W > ( ⎤ ⎦
W > x⎤ =
⎣
ln
⎣
⎛ ln ( x) −10 ⎞
= − Φ ⎜ ⎝ . 1 5 ⎟ ⎠ = . 0 999
1
2
From Appendix Table III, 1 − Φ( ) = .0 99z when z = − .33. Therefore,
ln x ( ) − 10
= − .33 and x = exp . (6 505 ) = 668 .48 hours
2
. 1 5
Determine the mean and standard deviation of lifetime. Now
2
E X ( ) = e θ+ω / 2 = e (10 + .125 ) = 67 ,846 .3
1
−
V X ( ) = e 2 θ+ω 2 ( e ω 2 −1 ) = e 20 ( + .225) (e 2 25 1 )
2
.
= 39 070 059 886 6 ., , ,
so the standard deviation of X is 197,661.5 hours.
Practical Interpretation: The standard deviation of a lognormal random variable can be large relative to the mean.
Exercises FOR SECTION 4-11
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion.
4-170. Suppose that X has a lognormal distribution with 4-173. The length of time (in seconds) that a user views
2
parameters θ = 5 and ω = 9. Determine the following: a page on a Web site before moving to another page is
(
(
(a) P X <13 ,300) (b) Value for x such that P X ≤ x) = .95 a lognormal random variable with parameters θ = .0 5 and
0
2
(c) Mean and variance of X ω = 1.
4-171. Suppose that X has a lognormal distribution with (a) What is the probability that a page is viewed for more than
2
parameters θ = − 2 and ω = 9. Determine the following: 10 seconds?
(
(
0
(a) P 500 < X <1000) (b)Valuefor x such that P X < x) = .1 (b) By what length of time have 50% of the users moved to
(c) Mean and variance of X another page?
4-172. Suppose that X has a lognormal distribution with (c) What are the mean and standard deviation of the time until
2
parameters θ = 2 and ω = 4. Determine the following in parts a user moves from the page?
(a) and (b): 4-174. Suppose that X has a lognormal distribution and
(
(a) P X < 500) that the mean and variance of X are 100 and 85,000, respec-
2
(b) Conditional probability that X <1500 given that X >1000 tively. Determine the parameters θ and ω of the lognormal dis-
ω
2
exp
(c) What does the difference between the probabilities in parts (a) tribution. [Hint: dei ne x = exp θ ( ) and y = ( ) and write
and (b) imply about lifetimes of lognormal random variables? two equations in terms of x and y.]