CEE_304_1st_Prelim2000 - CEE 304 UNCERTAINTY ANALYSIS IN...

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Unformatted text preview: CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING PRELIM #1 October 4, 2000 Use the text, notes, calculators, and your knowledge, to answer the questions below. The exam lasts 50 minutes. There are 50 points in total. 1) (6 pts) Consider two events denoted W and H where P[W] =0.6, P[H] =02, P[HU W] =r-0.7 (a) What is the probability of both H and W occurring? (b) What is the probability of W not occurring at the same time that H does occur? (c) If one has observed that H occurred, what is the probability W occurs? Show work. 2) (8 pts) A student is trying to assemble a flashlight becauSe of a power failure. Their younger brother took the three matching flashlights apart. There are now 3 bulbs, 3 flashlight bodies, and 5 batteries. Two of the three bulbs work, and 4 of the 5 batteries work. The bodies all work. If one bulb, one body, and 2 batteries are selected at random, what is the probability the flashlight when assembled will fail? How many different (unique) pairs of batteries can the student select? Show work. 3. (8 pts) Professor Stedinger is scoutmaster of Boy Scout troop 2 in Ithaca. The troop is planning its annual fall fund raising drive: selling popcorn. Suppose 2 patrols will in expectation sell $600 and $800 worth of popcorn, with standard deviations of $150 and $200, respectively. The correlations among the amounts each patrol sells is 0.60. What total value of popcorn (the sum of that sold by both patrols) can Scoutmaster Stedinger be 95% certain the patrols will sell. (Use a normal distribution.) 4. (13 pts) In the Western United States in August, major forest fires in a dry year start somewhere at the average rate of two every 5 days. (a) Briefly, why might a Poisson Process be a good model of the occurrence of such fires? (b) What is the probability a whole week (7 days) goes by without a fire starting? (c) What is the probability of having 2 or more fires begin between August 12 & August 18 ? (7 days) (d) Starting on August 2nd, what is the mean and standard deviation of the time until 8 fires start? 5. (8 pts) The probability density distribution function a random variables Q is: fQ(q) = =(3-q)/4 Osqu, 0 otherwise Let E = 00-5. What are the probability density function and the cdf for E? 6. (7 pts) Sally is in charge of checking the tires on the corporate fleet of SUVs. Careful inspection reveals that 60% of the FireBridge and 5% of the GoodMonth tires have problems, the two sources for tires. If one third of the tires are made by GoodMonth, then overall what fraction of the tires have problems? Across the entire fleet, what fraction of the tires with problems were made by FireBridge? CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING SOLUTIONS for Prelim #1 October 4, 2000 Axioms of Probability (Draw Venn diagram. Makes the problem easy. Many people missed key points.) 1a)P(WUH)=P(W)+P(H)-P[WflH] => P[WflH] =0.10 2pts 1b) P[W' 0H] =0.1 [Lookatdiagram] 2pts 01‘ P(H) = P(WflH) + P(W'flH) => 0.2 = 0.1 + P(W'flH) => P(W'flH) = 0.1 1c) P[ W l H] = P(WflH)/P(H) = 0.1/0.2 = 0.5 2 pts Counting (trouble here) 2a) Compute first probability of non—failure: Pr(Works) = (2/3) (1/1) (4/5) (3/4) = 2/5 = 0.4 [Drawing without replacement] 5 pts Hence Pr(Failues) = 1— Pr(Works) = 0.6 1 pts To work one must have one of 3 good lights, and must have selected first one of 4 good batteries from 5, and then one of 3 good batteries from remaning 4. Equivalently: Pr(Works) = C(2,1)C(3,1)C (4,2)/{C(3,1)C(3,1)C(5,2)} = 2/5 = 0.4 Note: (4/5) (3/4) = C(4,2)/C(5,2) = Perm(4,2)/Perm(5,2) = 3/5 2b) Combinations(5,2) = C(5,2) = 5*4/2 = 10. 2 pts Normal calculations 3a) S = P1 + P2. Sum of RVs. E[S] = 600 + 800 = 1,400 2 pts Var[S] = 1502 + 2002 + 2(0.60)(150)(200) = 98,500; StDev[S] = 313.9 3 pts 5005 = E[S] —— 1.645 Stdev[S] = 1400 — 1.645*313.9 = $884 3 pts Poisson process -- Gamma and Poisson distributions (Straightforward) 4a) Poisson process is reasonable because fires arrive separately, with p = hAt, we can assume that arrival rate 7» is constant across August, and beginning of fires in distant places is likely to be independent of one another, once we have conditioned upon it being a dry year in the West. 3 pts 4b) With )t = 2/5 ; Pr(K = 0 for t :7) = exp(-)tt) = 0.0608 3 pts 40) Pr(K 2 2) = 1—Pr(K = 0 or 1) = 1 - exp(-)tt) - (M) exp(-}tt)/1! = 1—0.06—0.17 = 0.770 3 pts 4d) Arrival time to 8th is gamma with a = 8, B = ll)» :25. y = 043 = 20 days; 02 = 0:02 = 50 ; o 2 St, [291 = LQZ days [Remember units] 4 pts Pdfs, Moments and Independence 5) Compute FQ(q) = (6q — q2)/8 for o s q s 2 3 pts. cdf:FE(e)=Pr{Ese}=Pr{Q0-5se}=Pr{Qse2}=(6e2-e4)/8, Oses 1.414 3pts pdf: fE(e) = (12c - 4e3)/8 = (3e - e3)/2 o s e s 1.414 [Range is important] 2 pts 0R: fE(e) = fQ( q[e] ) | dq/de | for l—to—l-monotone funcs => {(3 - e2)/4}*(2e) = (3e - e3)/2 Bayes Theorem / ---—Bad (0.60) / —-—Firebridge (2/3)~----0-----Okay (0.40) ——————— —-°Goodmonth (1/3) —-—0———-—Bad (0.05) \ —---Okay (0.95) 6) Fraction bad tires: P(Bad) =P(BTlFireBdge)*P(HreBdge) + P(BadIGoodmth) *P(Goodmth) = 0.60(2/3) + 0.05(1/3) = 0.4 + 0.01667 = 0.41667 ‘ 4 pts Bayes Theorem: PI Firebdge l Bad 1 = P(Bad l Firebdge)*P(Firebdge)/P(Bad) = 0.96 3 pts WS/ 2590 :0) Agfivm H m we .5 w mo wig ma H md o macros—E “£980 5:33—ch No E 0.0 W wd ...
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