210-kunen-07fafin

210-kunen-07fafin - Page 1 of 16 MATH 210 FINAL Semester 1,...

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Unformatted text preview: Page 1 of 16 MATH 210 FINAL Semester 1, 2007—2008 Lecture 1 Name: Circle your TA and section number on each sheet TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 ' sections Loizos Antonio M, 7:45 301 M, 8:50 303 W, 7:45 302 W, 8:50 304 M, 2:25 311 M, 9:55 305 W, 2:25 312 W, 9:55 306 Meghan NO CALCULATORS, NOTES, BOOKS, ETC. ALLOWED. EXPLAIN YOUR WORK. Unless you are instructed otherwise, your answer should be computed .com- _, I pletely (6.9., as a whole number, or a simple fraction, or a decimal). I, Numberl MAX Grade 1 7 2 9 3 8 4 10 5 10 6 10 7 8 8 7 9 7 10 7 11 10 12 7 LSUM L 100 j Page 2 of 16 1. (7 points) You roll a fair die three times, and you win if the three numbers you get are in strictly increasing order. So, (1,3,5) and (2,5,6) both win, but (2,2,4) and (4,3,2) both lose. Find the probability that you Win. Page 3 of 16 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 2. (9 points) Let Pr be a probability measure on S with E, F, G C 3. Assume a that Pr[E] = 6 Pr[F] = 7 Pr[G] = 8 Pr[EUF] = 11 Pr[EUG] = 10 ‘17? T77 "17’ 17> 1?, Pr[FU G] = 1%, and Pr[E‘fl Ffl G] = Find each of Pr[EU FU‘G] and Pr[E U F U G’]. Page 4 of 16 3. (8 points) Consider the following game: Start with a deck of six cards: the 2,3,4 of diamonds plus the 2,3,4 of clubs. You randomly choose two cards and you then get paid the product of the numbers on the two cards. So, if you draw the (>3, #3, you get $9, and if you draw the ()2, ()4, you get $8. Find the expected value of your payoff. Page 5 of 16 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 4. (10 points) Consider the following game: You toss four fair coins — so you get between 0 and 4 heads. If you get 3 or 4 heads, you get paid $16. If you get 2 heads, you get paid $8. If you get 0 heads or 1 head, you get paid nothing. Find the expected value, the variance, and the standard deviation of the amount of money you get paid. You may leave the result for 0(X) in terms ofax/ . Page 6 of 16 5. (10 points) Solve the system of equations: , 7:0 + By + 32 15 3:3 + By + 72 = 9 5:1: + 10y + 152 20 There is only one solution. . Page 7 of 16 Name: W TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 6. (10 points) A Markov chain has three states: State 1, State 2, State 3. It has the transition matrix P shown below. Find the vector W of stable probabilities 1 (so, 14/13 2 14/ There is a unique solution here. l0de Ali—Wm .5 P: .3 .1 £2 Pagefl’ of 16 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 7. (8 points) A fair die is rolled 18,000 times. Use the normal approximation _ to the binomial to estimate the probabilities of the following happening: a. You roll the number one at least 2,950 times. b. The number of ones that you roll is between 2900 and 3015. The next page has a table of the areas under the standard normal curve. ‘1 Pageflof 16 Area under the Standard Normal Curve .00 r .01 .02 .03 .04 .05 .06 .07 .08 .09_+ 9 2 0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .035 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 -.1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .2257 .2291 .2324 .2357 v .2389 .2422 .2454 .2486 .2517 .2549 0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 . .4192 .4207 .4222 .4236 .4251 .4265; .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357 .4370. .4382 .4394 .4406 8 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 :4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986 3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990_, 10 (a Page/1’1 of 16 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 8. (7 points) Let X be a random variable With probability density function f , where 0 if cc < —3 0.1 if—3£x<3 0.2 if3§$<5 O if5§a§ Find Pr[X 2 —2 I X < 4] and Pi"[X S 4 | X Z —2]. f(x) = 11 ‘U Page )2 of 16 9. (7 points) A five card poker hand is dealt at random from a standard deck of cards. Find the probability that the number of clubs is at least two- more than the number of diamonds (for example, one diamond and four Clubs, or zero diamonds and two clubs, etc). Here, you may leave your answer as an arithmetical expression, without evaluating it 12 (L. Page 1/3, of 16 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 10. (7 points) Suppose that you take out a 20 year mortgage for $200,000 at 12% annual interest rate; so you pay it back monthly over the next 20 years. Assume that the loan is amortized in the standard way. Find your monthly pay- ments. Here, you may leave your answer as an arithmetical expression, without evaluating it. 13 (3 Page Met" 16 11. (10 points) Find the maximum and minimum values of :r + y subject to the constraints: 4$+5y£20 5x—y25 5x—3y315 If the maximum and/ or minimum doesn’t exist, say so. There’s some graph paper on the next page. 14 Name: TA: Loizos Antonio Meghan Section: 301 302 311 312 303 304 305 306 307 308 309 310 15 :3 Page Moi 16 12. (7 points) You need to have some money thirty years from now, so you put money in a sinking fund, depositing monthly into an account which pays 3% interest (annual rate), compounded monthly. Each month you deposit $200. How much money will you have at the end of the thirty years? Here, you may leave your'answer as an arithmetical expression, without evaluating it. 16 ...
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210-kunen-07fafin - Page 1 of 16 MATH 210 FINAL Semester 1,...

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