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260
CHAPTER 4
Commonly Used Distributions
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FIGURE 4.16 (a) A hi stogram showing monthly production fo r 255 gas wells. There is a lo ng righth and tail. (b) A histogram showing the natural logs of the monthly productions . The distribution of the logged data is mu ch closer to norma l.
Exercises for Section 4 .61. The life ti me (in days) of a certai n electron ic compo nent that operates in a hi g htemperature environment is lognonnally di stributed with /J = 1.2 and a = 0.4. a. Find the mean lifetime. b. Find the probability th at a component lasts be tween thre e and six day s. c. Find the median lifetime. d . Find the 90th percentile of the lifetimes. 2. The article "Assessment of D ermophaImacokinetic A pproach in the Bioequivalence Determinati on of Topical Tretinoin Gel Products" (L. Pershing, J. Nelson, et a1.. JAm Acad Dermatol 2003:740751) reports that the amount of a celtain antifu ngal oin t ment th at is absorbed into the skin can be modeled with a lognormal di stribution. As sume that the amount (in ng/cm2 ) of acti ve ingredien t in the ski n two hours after app lication is log normally distributed with /J = 2.2 and a = 2.1. a. Find the mean amount absorbed. b. Find the median amount absorbed . c. Find the probability that the amount absorbed is more th an 100 ng/cm 2 . d. Find the probability that the amou nt absorbed is less than 50 ng/cm2 . e. Find the 80th percentile of the amount absorbed. f. Find the sta ndard deviation of the amount absorbed. 3. The body mass index (BMI) of a person is define to be the perso n's body mass divided by the sq uar of the person's height. The article "Influences 0 Parameter Uncertainties within the ICRP 66 Respi ratory Tract Model: Particle Deposition" ( w. Bol d E. Farfan, et aI. , Health Physics, 2001 :378394) stat that body mass index (in kg/m2) in men aged 2534 i log nonnally distributed w ith parameters Jl = 3.2 1.' and a = 0. 157. a. Find th e mean BMI for men aged 2534. b. Find the standard deviatio n of BMI for men ab 2534. c. Find the median BMI for men aged 2534. d. What proportion of men aged 2534 have a B. less than 22? e. Find the 75t h percenti le of BMI for men 25 34 .a ~_
4.6
The Lognormal Distribution
261
The article "Stochastic Estimates of Exposure and Cancer Risk from Carbon Tetrachloride Released to the Air from the Rocky Flats Plant" (A. Rood, P. McGavran , et a!. , Risk Analysis, 2001:675695) models the increase in the risk of cancer due to exposure to carbon tetrachloride as lognormal with IL = 15.65 and (j = 0.79. a. Find the mean risk. b. Find the median risk. c. Find the standard deviation of the risk. d. Find the 5th percentile. e. Find the 95th percentile. 5. If a resistor with resistance R ohms carries a current of I amperes, the potential difference across the re sistor, in volts, is given by V = JR. Suppose that J is lognormal with parameters ILl = 1 and (j; = 0.2, R is lognormal with parameters ILR = 4 and (j~ = 0.1, and that I and R are independent. a. Show that V is 10gnonl1ally distributed, and com pute the parameters f.tv and (j~. (Him: In V = In I + In R.) b. Find P(V < 200). c. Find d. Find e. Find f. Find g. Find h. FindP(150 ::: V ::: 300). the mean of V. the median of V. the standard deviation of V. the 10th percentile of V. the 90th percentile of V.
b. What is the mean withdrawal strength for helically threaded nails? c. For which type of nail is it more probable that the withdrawal strength will be greater than 50 N/mm? d. What is the probability that a helically threaded nail will have a greater withdrawal strength than the median for annularly threaded nails? e. An experiment is performed in which withdrawal strengths are measured for several nails of both types. One nail is recorded as having a withdrawal strength of 20 Nlmm, but its type is not given. Do you think it was an annularly threaded nail or a helically threaded nail? Why? How sure are you?8. Choose the best answer, and explain. If X is a random variable with a lognormal distribution , then _ _ __
i. the mean of X is always greater than the median.II. Ill.
the mean of X is always less than the median. the mean may be greater than, less than, or equal to the median, depending on the value of (j.
9. The prices of stocks or other financial instruments are often modeled with a lognormal distribution. An in vestor is considering purchasing stock in one of two companies, A or B. The price of a share of stock today is $ 1 for both companies. For company A, the value of the stock one year from now is modeled as lognormal with parameters f.t = 0.05 and (j = 0.1. For com pany B, the value of the stock one year from now is modeled as lognormal with parameters IL = 0.02 and (j = 0.2.a. Find the mean of the price of one share of company A one year from now. b. Find the probability that the price of one share of company A one year from now will be greater than $1.20. c. Find the mean of the price of one share of company B one year from now. d. Find the probability that the price of one share of company B one year from now will be greater than $1.20.10. A manufacturer claims that the tensile strength of a certain composite (in MPa) has the lognormal distli bution with IL = 5 and (j = 0.5. Let X be the strength of a randomly sampled specimen of this composite.
6. Refer to Exercise 5. Suppose 10 circuits are con structed. Find the probability that 8 or more have volt ages less than 200 volts. 7. The article "Withdrawal Strength of Threaded Nails" (D. Rammer, S. Winistorfer, and D. Bender, Journal of Structural Engineering 200 I :442449) describes an experiment comparing the ultimate withdrawal strengths (in N/mm) for several types of nails. For an annularly threaded nail with shank diameter 3.76 mm driven into sprucepinefir lumber, the ultimate withdrawal strength was modeled as lognormal with IL = 3.82and(j = 0.219. Fora helically threaded nail under the same conditions, the strength was modeled as lognormal with f.t = 3.47 and (j = 0.272. a. What is the mean withdrawal strength for annu lady threaded nails?
a. If the claim is true, what is P(X < 20n
262
CHAPTER 4
Commonly Used Distributions
b. Based on the answer to part (a), if the claim is true, would a strength of 20 MPa be unusually small? c. If you observed a tensile strength of20 MPa, would this be convincing evidence that the claim is false? Explain. d. If the claim is true, what is P(X < 130)7 e. Based on the answer to part (d), if the claim is true, would a strength of 130 MPa be unusually small?
f. If you observed a tensile strength of 130 MPa, would this be convincing evidence that the claim is false? Explain.
11. Let XI,"" X" be independent lognormal random variables and let a, . .... a" be constants. Show that the product P = X~' X;;" is lognormal. (Hint: In P = a, In X I + ... + an In X,,).
4.1 The Exponential DistributionThe exponential distribution is a continuous distribution that is sometimes used to model the time that elapses before an event occurs. Such a time is often called a waiting time. The exponential distribution is sometimes used to model the lifetime of a compo nent. In addition, there is a close connection between the exponential distribution and the Poisson distribution. The probability density function of the exponential distribution involves a parameter, which is a positive constant A whose value determines the density function's location and shape.
Definition
'.
,
The probability density function of the exponential distribution with parameter A > 0 is Ax x> 0 f(x) _ {Ae (4.32) 0
xsO
Figure 4.17 presents the probability density function of the exponential distribution fo: various values of A. If X is a random variable whose distribution is exponential wi parameter )c, we write X ~ EXp(A). The cumulative distribution function of the exponential distlibution is easy to com pute. For x < 0, F(x) = P(X S x) = O. For x > 0, the cumulative distributi function isF(x) = P(X S x) =
l'
Ae
Xl
dl = 1  e h
SummaryIf X~
.
. '
,
EXp(A), the cumulative distribution function of X is I Xx x> 0 F(x) = P(X S x) = 0  e { XSO
(4.33)