Final Exam

Final Exam - MATH 3C (Spring 2007, Lecture 2) Instructor:...

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Unformatted text preview: MATH 3C (Spring 2007, Lecture 2) Instructor: Roberto Schenrnenn Final Exam Last Name: First and Middle Names: Signature: UCLA id number (if you are an extension student, say so): Circle the discussion section in Which you are enrolled: 2A (T 10F Akemi Kashiwada) 2B (R 19; Akemi Kashiwarla) 2C (T E0, Wenhua Geo) 29 (R 10, Weuhua. Geo) 213 (T 10, llhwan Jo) 2F (R El}, lllzwan $50) When the instructions to a question ask you to explain your zusswer7 yeu should show your work and explain whet you are doing carefully; this is then more important than just finéing the right answer. Please, write clearly and make Clear what your solution and answer to each problem is. When you continue on another page indicate this clearly. To cancel anything from your solution, erase it or cross it out. You are not allowed to sit Close to students With Whom you have studied for this exam, or to your friencis. Enjoy the exam, and Good Luck 1 1) (10 points) Roll :1 die 2 times. What. is 1ighe condiijmzal probability that: the sum of the faces shown is; 6? given Ehat at least one of the faces shown is a 4? (Give answer as a. fracéion or in decimal form.) (Show and explain your wark.) .w’“ 1-K" . 9n ' g W"? 3W, 3 saw" m; 3 : i E 1 E é “a! ' \ y“ “:23; g "a? r \ A“? a .55" i .x' . mg" I,» '53 2) (10 peims) In how many ways: can we give 3 prizes to a gmup of 190 people; if each prize can oniy be given to (me person; and each person can receive a? most. one prize. (vaide a. numerical answer.) (No expianation needed, 312m; the answer is enough.) 3) (E0 points) You are dealt 6 cards from a standard deck of 52 cards. Con:- pute the probabiiity tho: among; those cards you have: 3 cards of one denomi— nation and 3 cards of another denomination. (No need to compute factorials, powers, permutations and combinations.) (No explanation needed? just. the answer is eaough.) 4) ($0 points) R01} 3. fair die. Consiéer the events- A = {1; 3}, B m {3,4, Are A and B independent? (Explain your answer carefuily.) {‘x g‘ 3;: my» 3?} w wm E 2 ’ 2 4‘ W/ / Km 1;: Wt. . V . .4 - .p a w « x, «x 1 w .v 5) (10 points) A greup of people has '20 adults and 20 children. One of {he adults- is- called Ann and another one Bob. We want te select 3 adults and '7 Children fio go on a. trip. They will go in a vehicle with room for £0, and one of the selected adults has te be the driver. (All {he adults are assumed to be potential drivers). Suppose that any possible choice of driver and passengers is equally likely. What. is the probebilitv that Ann will he the driver and Bob will be a passenger (not the driverfi (Provide a numerical answez es a fractioa or in decimal farm.) (Show and explain your work.) "28%. “a. 6) (10 points) A screening test for a. disease shows a. false positive with probability 1% and a false negative with probabiiity 2%. In the population 03% of people have thee disease. Given. that someone tested positive for the disease, what is $118 probability that heshe has the disease? (Provide a numerical answer as a. fraction or in decimal form.) (Show and expiain your work.) 7’) (10 points) You have three boxes. The first box contains 200 blue bails and 100 yeiiow baiis. The second box contains 200 blue belie and 200 yeilow balis. The shird box oon’sains 200 biiie bells aged 300 yeliow bails. A box is seieoted at random; and from this box balls are selected Without replaoeniens. Let. S be the random variable ihet gives the nurnfier of yellow bails among {he selected ones. Find P(S§1). (No need to compute feefioriais, powers, permutations and combinations) (No explanation needed7 just the answer is enough.) 8) (10 points) Suppose them: the independent events A and B have probabil— ities PUD m .6 and. PUB) m Compute PTA U B). (Provide a numerical answer in decimal} form.) (Show and expiain your work.) m wig/f Wm iv; a - 3 s, f :7 g 03‘ fl, :q a" m» g a} fi v s. F" A: g A” w w v M h 50 W» .fi Z X In; fl 9) (10 points) The mean deviation of a discrete raziéom variable X is defined as Zn m th grain)a Where pxm) : P(X m n) is the probability mass function of X 7, and the sum is over all the values that X can take. Compute the mean deviation 01" a random variable X that has binomial distribution with 4 attempts and probability E /2 of success each time; fie, X N BM; 1 /2). (Give answer as a fraction or in decimal form.) (Show and explain your work.) 10) (10 points) Suppase that X and Y are independent random variables with pmbabiiity 11121.55 funciiions given by: pX(-—1) = .3, pg; (0) mm: .4, pX(1) m .2, MO) = ,12 and pyfi) m .2, py(2) m .8. Compute P(X ~i— Y 2 2). (Give answer in (iecimal form.) (Show and explain your work.) ,, 5 fax g {jgxgysi 33$ “35w e A w- _ m ‘ ’x 5‘ v;- E-Jfli' e M" 5:» W 3 w‘ ‘ .M w ‘3 F ‘u w r 32; z ~ 1/ w = ¢~._ w (5 gum " vi 1.4 ii) (10 points) When Helen tosses a. dart at a target, the (Eistance (in inches) from the center of the target that she hits is we]? éescribed as a. continuous random variable X with probability densityr functien given below, where C’ is an appropriate number. 0 if s<0 ors>§0, XS ‘” 05 if ogsgie. 21) Find the value of C. b) If she tosses the dart at the target 5 times, what is the probability that at least once the dart hits the target less than 2 inches from the center? Assume independence of the 5 tosses. (Provide numerical answers in éecimel form.) (Show and explain your work.) 12) (10 points) A cantinuous random variable Y has probability density func— tien given below. 0 if s<——1ors>l. £43 = 1+5 if-—1§5<:Q lws if OSSSl. Compute VarCY). (Provide a numerical answer in decimal form.) (Show and {explain your work.) éggfi} é :2 % nglu _ “MWAWWMWWWMT / © a: 2;, ’1 ( g \ ~ ‘7 k “l m 1‘ E Z 3 {:2 J‘J“: xr-A a \_ a {if g z / ¥ 1 l l: 2; ‘ l i \ 1 a z \ E Styli}: £1:( '3: l ’36?» 55;? + {Edi/3:; : u \j g g fig 7%” 3’ S we FEE: ff? 3:93 * . a 2 xgx‘gig :: «9r “Wig : {WT/Escld-M ‘xil3‘6“’3’9 "’ “l 3:: J" w’ < «bf g ' «a w, C} a § , m m"; '7 f if s 371 l w 4"“: w .4,» M “@wlflilélsl u~-wéwl>a : fl» / W» "Mi" wwjgfifi W .i «2 é 1am ? x 2 x” v 3" p g a gig/19¢ Véq/ (y; a: f 26% {é} “ii” “" f f, I"; W“ M 1.? 1w :“‘”~’&5§f“ m >’ g j 13) (10 points) Suppose mat. W is a yandom variable with normal distribution with mean ——1 and stanéard deviation 2. Compute P(—3 < W < 415). (Provide a. Immez‘ical answer in decimal form.) (Show and explain your work.) L t " mm. M. Kim mm, {Ha 14-) (10 points) A certain substance in the blood is distributed as a normal random vari able with mean 290 and standard deviation 10 (when measured in rng/dl). People who have an amount larger than 225 nag/d} are diagnosed as having a certain medical condition. When the amount is larger than 215 mg/ (11, doctors call the patient (If the amount is between 215 ng/ CH and 225 mg/dl they recommend life style changes, to prevent the condition from developing; if the amount is larger than 225 rug/(ll they start treating the condition with medications; but if the amount is less than 21-5 :ng/ (:11 they do not even call the patient.) If after having a sample of your blood taken to check on that substance you receive a phone call from your doctor, what is the conditional probability that you have the medical condition? (Provide a numerical answer in decimal form.) (Show and explain your work.) L5) (10 points) Toss a. fair coin :00 times. Use the central limit theorem to find an approximation for {he probabiiity that the number of heads is at most '55. (Provide a numerical answer in decimal} form.) (Shaw and explain your work.) 'i 6 ‘ (5%: M é «Law-- I “<1 W 5’“ g‘ g 9»; Rx? W 3' 4‘3 z .MJ—v“ f— 8 E Eable of the Siandard Nommi Distribution 919 TABLE OF THE STANDARD NORMAL QESTRIBUTION Areas under the Standard Normai Curve from ~00 :0 2: (see Figure 8.1}. 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5754 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141 0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 0.6 .7258 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7518 .7549 0.7 .7580 .7612 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 0.8 .7881 .7910 .7939 .7967 .7996 .8023 .8051 .8078 .8106 .8133 0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 W 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .991 .9913 .9916 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.6 .9953 .9955 .9955 19957 .9959 .9960 .9951 .9952 .9963 .9964 2.7 .9965 .9966 .9957 .9958 .9969 .9979 .9971 .9972 .9973 .9974 2.3 .9973 .9975 .9976 .997? .9977 .9978 .9972: 599%; .9980 .9981 2.9 .9981 .9932 .9982 .9983 .9984 .9934 .9935 .9935 .9985 .9986 ...
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Final Exam - MATH 3C (Spring 2007, Lecture 2) Instructor:...

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