Examination Midterm # 1
Sta 113: Probability and Statistics in Engineering Thursday, 2004 Sep. 30, 1:15 2:30 pm
This is a closed-book exam so do not refer to your notes, the text, or any other books (please put them on the floor). You may use a single sheet of notes or formulas and a calculator, but materials may not be shared. Normal Z, a p.d.f. formula sheet, and two blank worksheets are attached to the exam. You must show your work to get partial credit. Even correct answers will not receive full credit without justification. Please give all numerical answers to at least four correct digits or as exact fractions reduced to lowest terms. Write your solutions as clearly as possible and make sure it's easy to find your answers (circle them if necessary), since you will not receive credit for work that I cannot understand or find. Good Luck! If you find a question confusing please ask me or Floyd to clarify it. Cheating on exams is a breach of trust with classmates and faculty, and will not be tolerated. After completing the exam please acknowledge the Duke Honor Code with your signature below: I have neither given nor received unauthorized aid on this exam.
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Problem 1 : Let A and B denote two events in a probability space. Decide if the following statements are true or false for any A and B and explain WHY. (15pt) i If A and B are independent, then P (A) + P (B) = P (A B) ii If A and B are independent, P (A) = 0.3 and P (B) = 0.2, then A and B are not mutually exclusive. iii If A and B are mutually exclusive, then P (Ac B c ) = P (Ac ) + P (B c ) iv P (A) = P (A B c ) + P (A B) v P (A|B) P (A B)
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Problem 2 : A box in a certain supply room contains six 40-W lightbulbs, four 60-W lightbulbs, and two 75-W lightbulbs. Suppose that three bulbs are randomly selected. Which is more likely: that all three have the same rating , or they have all different ratings? Show your work (which include computation of both probabilities). (15pt)
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Problem 3 : Components of a certain type are shipped to a supplier in batches of ten. Suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected and tested. a. (15pt) What is the probability that neither component is defective? b. (5pt) If neither component is defective, what is the probability that the batch contains two defective components?
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Problem 4 : A reservation service employs five information operators who receive requests for information independent of one another, each according to a Poisson process. Suppose the mean time between requests for one operator is 30 seconds. What is the expected number of operators who receive no requests during a given 1-min period? (10pt)
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Problem 5 : . Volkswagen speedometers give exactly the correct speed on average, but with a normally-distributed error having standard deviation 5mph; thus the true speed of a driver whose speedometer reads s will really be a random variable X with a normal distribution having mean s and variance 52 = 25. Val is driving a Volkswagon on Route I-40, where the posted speed-limit is 65mph. Val is in a hurry, but knows there is a speed trap along this route; in North Carolina, State Troopers issue tickets to all drivers exceeding the posted speed limit by at least 10mph. a. (10pt) If Val is driving at the speed of 70mph as shown on the speedometer (i.e., the true speed is a normal random variable with mean 70mph and variance 25). What is Val's chance of receiving a ticket? b. (10pt) What is the fastest Val can try to drive (i.e., what will the speedometer say) and still have only a 1% chance of receiving a ticket?
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Problem 6 : figure below.
Consider a system consisting of two components as in the
1
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The system works if either 1 or 2 works. Suppose the lifetime of each component, measured in hours, is a random variable Xi , i = 1, 2, with density function fXi (x) = where 1 = 1/2 and 2 = 2/3. a. (5pt) Calculate the expected lifetime of each component. b. (15pt) Let X denote the life time of the whole system (measured in hours). Find density the function of X. i e-i x , if x 0; 0, if x < 0.
2
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(Nearly) Blank Work-Sheet #1 (of 2)
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(Nearly) Blank Work-Sheet #2 (of 2)
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z
(z) =
-
1 2 e-t /2 dt: 2
.02 .5080 .5478 .5871 .6255 .6628 .6985 .7324 .7642 .7939 .8212 .8461 .8686 .8888 .9066 .9222 .9357 .9474 .9573 .9656 .9726 .9783 .9830 .9868 .9898 .9922 .9941 .9956 .9967 .9976 .9982 .9987 .9991 .9994 .9995 .9997 .03 .5120 .5517 .5910 .6293 .6664 .7019 .7357 .7673 .7967 .8238 .8485 .8708 .8907 .9082 .9236 .9370 .9484 .9582 .9664 .9732 .9788 .9834 .9871 .9901 .9925 .9943 .9957 .9968 .9977 .9983 .9988 .9991 .9994 .9996 .9997
.......... . ............ ... .. ... ... . ... .. .. . . . . . .... .. . .. .. .. .. . . . . . . . . ... . .. . .. .. . . . . . . . ... .. .. . .... . . . .. . . .. .. .. . . . . . . . . . . . . . . .... . .. .. .. . . . . . . . . . . .... .. .. .. . . . .... . . .. ... . . . . . ... ... .... . . . . . . . . . . . . . . . . . . . . . ...... .... . .. ..... ....... .... .... . . . . . . . . . . . . . . .. .. ................ ............ ................ ................
(z)
9
-3 -2 -1 .05 .5199 .5596 .5987 .6368 .6736 .7088 .7422 .7734 .8023 .8289 .8531 .8749 .8944 .9115 .9265 .9394 .9505 .9599 .9678 .9744 .9798 .9842 .9878 .9906 .9929 .9946 .9960 .9970 .9978 .9984 .9989 .9992 .9994 .9996 .9997
0
1 z 2
3
Table 4 z .00 0.0 .5000 0.1 .5398 0.2 .5793 0.3 .6179 0.4 .6554 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 .6915 .7257 .7580 .7881 .8159 .8413 .8643 .8849 .9032 .9192 .9332 .9452 .9554 .9641 .9713 .9772 .9821 .9861 .9893 .9918 .9938 .9953 .9965 .9974 .9981 .9987 .9990 .9993 .9995 .9997
.01 .5040 .5438 .5832 .6217 .6591 .6950 .7291 .7611 .7910 .8186 .8438 .8665 .8869 .9049 .9207 .9345 .9463 .9564 .9649 .9719 .9778 .9826 .9864 .9896 .9920 .9940 .9955 .9966 .9975 .9982 .9987 .9991 .9993 .9995 .9997
.04 .5160 .5557 .5948 .6331 .6700 .7054 .7389 .7704 .7995 .8264 .8508 .8729 .8925 .9099 .9251 .9382 .9495 .9591 .9671 .9738 .9793 .9838 .9875 .9904 .9927 .9945 .9959 .9969 .9977 .9984 .9988 .9992 .9994 .9996 .9997
CDF for Standard Normal. .06 .07 .08 .09 .5239 .5279 .5319 .5359 .5636 .5675 .5714 .5753 .6026 .6064 .6103 .6141 .6406 .6443 .6480 .6517 .6772 .6808 .6844 .6879 .7123 .7454 .7764 .8051 .8315 .8554 .8770 .8962 .9131 .9279 .9406 .9515 .9608 .9686 .9750 .9803 .9846 .9881 .9909 .9931 .9948 .9961 .9971 .9979 .9985 .9989 .9992 .9994 .9996 .9997 .7157 .7486 .7794 .8078 .8340 .8577 .8790 .8980 .9147 .9292 .9418 .9525 .9616 .9693 .9756 .9808 .9850 ....