304 CEE - UNCERTAINTY ANALYSIS IN ENGINEERING
Homework #7 Due: Due Monday, October 15, 2007. Read: Lec. Notes 18 and EXAMPLES! Text Section 4.5 (particularly Weibull and Lognormal discussion). By Friday, October 12, read sections 6.1 and 6.2 on Statistics, and Devore7 problem 34 on p. 252 [D6 279] for definition of MSE. Goal: This homework finishes our study of probability its applications. Statistics is the next topic.
What we have learned of probability allows us to tackle some interesting problems. This assignment provides an opportunity to learn about and use the lognormal, Gumbel, and Weibull distributions. The lognormal model is used as a model of a wide range of phenomena, both physical and biological. The Gumbel and Weibull distributions are especially applicable for the description of large and small extremes, as illustrated by problems 3 and 4 below. Problem 5 reviews the 2-parameter continuous distributions that have been covered in the course. You can use Excel to do computations, but be clear as to what function in Excel you are using and what the needed inputs are.
Assignments ActivStats (REQUIRED)
(a) Look at first 2 activities on page 17.1 of Activstats, called either: "Consider what we can learn..." (with 3 T/F question).........or......... 'Confidence Intervals and Sampling Distributions' The second lesson should be: "Understanding the confidence interval as balance..." .........or......... "Can We Estimate a Parameter?" (with 3 T/F questions) What are the answers to the 3 True/False questions? 3 T/F Questions Are ...Does sample mean equal the population mean? ...Does the sample mean probably equal the population mean? ...Can we say the sample mean +/- margin of error equals the population mean? [Plus: This had better work....] (b) Look at the activity on page 17.4, "Understanding Reasoning for Confidence Intervals?".........or......... "The Reasoning of Confidence Intervals?" What is the 95.4% confidence interval for height of British women?
Devore: (REQUIRED): Continuous Distributions:
1. A structural engineer believes the yield stress S in mild steel reinforcing bars to be lognormally distributed with E[lnS] = ln(40 ksi) ; Var[lnS] = (0.1) 2. The bar areas A of nominal 9/8 inch diameter bars ("No. 9 bars") are lognormally distributed with
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING
E[A] = 1.0 in2; Var(A) = [0.05 in2] 2 a) Assuming S and A are independent, find the distribution of the yield force: F = S . A b) What is the expected value of F? c) What is P[F < 35 kips]? {Review lognormal structural reliability example in Lec. Notes #16 for an example of how this problem can be solved.} 2. Consider a lognormal distribution for X: ln X = L ~ N( L, 2L). How is L related to the median of X's distribution? {Very simple relationship: see Lec Notes 16: with Yp~N, Y = Y + Y Zp ; Devore7, p. 169, #80a; p. 166// D6, p. 188, #74a} 3. Suppose the strength of a long cable of length n can be described by a Weibull distribution with k = 3 and u = 5. a) What would be the parameters of the Weibull distribution describing the strength of similar cables of length 5n? b) What is the ratio of the mean strength of cables of length n to that of cables of length 5n ? { Use properties for Weibull distribution in Lec Notes 18.} 4. Let X equal the velocity of the maximum wind guest in a day. Assume that X has an exponential distribution mean with 1/ . a) What is the mean and variance of X? What is FX(x)? b) Let W be the maximum gust that occurs in a year consisting of 365 independent days. Use a Gumbel distribution as an approximation of the distribution of W. (See Lecture Notes #18; just add to derivation there.) What is the approximate Gumbel cdf FW(w) for W? c) Using the Gumbel distribution approximation, what is the mean and variance of W as a function of for n = 365?
5. Several probability distributions have been covered this semester. Suppose the mean and variance of a random variable are 2 = 10.73 = 6.267 and Assume that the random variable has a (a) Gamma dist. [Excel has p = GAMMADIST(x,
,true);
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING
true returns cdf; false returns pdf. x p = GAMMAINV(p ) returns inverse of cdf]
(b) Normal dist. [Excel has p = NORMDIST(x, , ,true) & its inverse xp = NORMINV(p, , )] (c) Lognormal dist. [See Excel Normal functions, or for CDF use LOGNORMDIST, whereas
for inverse of CDF use LOGINV. Excel makes it easy.]
(d) Gumbel dist. [Excel has logs & exponentials. CDF and inverse CDF are closed form expressions.] (e) Weibull distribution; for the Weibull distribution the shape parameter k has a value of 2; remember that (1.5) = Then (Using spreadsheet; See examples with Lec. notes 18.): (i) What are the five distributions' parameters? (ii) What are the 1, 50, and 99 percentiles of the distributions? (iii) Provide a rough sketch of the shape of the pdf of each distribution. Make sure to get the shape right at x = 0 [A simple hand drawn sketch with the right shape is sufficient.] Answers
1. a) F ~ lognormal when ln(F) ~ N[ 3.688, (0.112)2] b) E[F] = E(S)E(A) or from distribution parameters = exp( + 0.5 2) = 40.20 c) P[F 35] = 0.12 2. How do quantiles of X relate to quantiles of L? 3. New u = 2.92 but k still equals 3. Shorter cable has a mean which is 1.7 times stronger. Value of k is critical to this calculation, and strength of long cables. 4. a) X ~ exponential( ) --> E(X) = 1/ ; Var(X) = 1/ 2 wln(n)] b) How does G(w) exp{ e [ } change when 1? While some students may feel that this problem represents theory, a very valuable insight is to understand how the distribution of a design variable like W would change as n increases. The Gumbel and Weibull distributions are examples of what we can understand of civil engineering system components by building probability models of how they behave. c) E[X] = 6.48/ ; Var[X] = 1.645/ 2 5. PARAMETERS <Remember to show sufficient work that we can grade homework.> a) = 3.661; = 1.712 b) same as moments c) = 1.715; = 0.491 d) = 0.3915; u = 4.79 e) k = 2 (k-value given because gamma functions are unpleasant); u = 7.07 Quantiles [to only one decimal place, you should supply another] Percent 1 10 50 90 Gamma 1.2 2.6 5.7 10.7 N LN 1.8 3.0 5.6 10.4 Gumbel 0.9 2.7 5.7 10.5 Weibull 0.7 2.3 5.9 10.7
/2 .
[Excel has GAMMALN(X) = ln (X) and FW(w) = WEIBULL(x, =k, =u true) - false yields pdf.]
-1.4
2.1
6.3
10.5
CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING
99 16.3 13.9
17.4
16.5
15.2
Note that 50 and 90 percentiles are very close, but 1 and 99 percentiles vary widely. WHY? From your knowledge of the shape of the pdfs, can you explain to yourself the reason for the differences in the observed quantiles? Do the results make sense? Why are medians of all of the distributions less than the mean, except for the normal distribution ?