PRACTICE PROBLEMS FOR THE FINAL Math 36Q Sprig 25 Professor Hoh Below is a list of practice questios for the Fial Exam. I would suggest also goig over the practice problems ad exams for Exam ad Exam 2 to help you prepare. Ay quiz, homework, or example problem has a chace of beig o the exam. For more practice, I suggest you work through the review questios at the ed of each chapter as well.. The amout of time a customer speds at a certai store is modeled by a expoetial radom variable with mea miutes. If each customer s time at the store is idepedet, use the Cetral Limit Theorem to approximate the probability that radomly selected customers sped betwee 95 ad 5 miutes at the store. Leave your aswer i terms of the stadard ormal distributio Φ(x). 2. Let X ad Y be joitly cotiuous radom variables with joit desity fuctio kx 2 y < x < y < f X,Y (x, y) = (a) Fid k. (b) Fid Cov(X, Y ). 3. This problem shows that ot all fuctios ca be momet geeratig fuctios, eve if they re very differetiable ear zero! Let f(t) = e t + t. (a) Preted that f(t) is the momet geeratig fuctio for some radom variable X ad use it to fid Var(X). (b) Usig the value of Var(X) from (a), how do you kow that o such X ca exist? 4. Suppose the umber of meteors hittig earth is modeled by a Poisso process (N t ) t with parameter λ = 5 (meteors / hour) where t is i hours. (a) What is the probability that durig the hour you are i this class, 2 or more meteors hit the earth? I do t wat you to leave me with a ifiite sum, istead it should be somethig that you could calculate easily if you were allowed a calculator. (b) What is the coditioal probability that durig the hour you are i this class, 2 or more meteors hit the earth give that or more meteors hit the earth? 5. Suppose X is a cotiuous radom variable with distributio fuctio give by x < F X (x) = x 2 x < x
(a) Fid the desity f X (x) of X. (b) Fid Var(X). 6. The followig parts (a) ad (b) refer to the letters MMMMPPIII. (a) How may words ca be created such that both P s occur to the left of the first I? (b) Suppose each word is equally likely to occur. What is the probability that you radomly select a word where both P s occur to the left of the first I? 7. Suppose that there is a disease withi a populatio. Each idividual withi the populatio idepedetly ad radomly has the disease with probability p. A certai test is created to scree idividuals for the disease. If a idividual has the disease, there is a probability q that the test correctly reports positive. If a idividual does ot have the disease, there is a probability r that the test falsely reports positive for the disease. If a radomly selected idividual tested positive for the disease, what is the probability that they have the disease? You should assume that the test oly reports a positive or egative result, ad your aswer should be i terms of p, q, ad r. 8. Your fried wats to play a gamblig game with you. The game goes as follows: You pay your fried $C to play. You the draw twice from a ur iitially cotaiig 4 red marbles ad 5 black marbles. O the first draw you take a sigle marble ad retur it back ito the ur with aother marble of the same color. O the secod draw you pull out a sigle marble: If it is a red marble you wi $5, whereas if it is a black marble you wi othig. Questio: What is the maximum amout of moey $C you should pay your fried to play so that o average you do t lose moey? 9. You must select exactly oe of two challeges: A or B. If you select challege A you are forced to aswer a questio which you have a /2 probability of gettig correct. If you aswer correctly you will be rewarded silver. If you aswer icorrectly, you will be forced to eat spoiled cheese. If you select challege B, you will be forced to complete a physical task which you have /3 chace of completig. If you succeed you will rewarded gold. If you do ot succeed you will be forced to eat spoiled cheese. You decide to flip a biased coi with probability 2/3 of gettig heads to decide which challege to take: heads meas you take A, tails meas you take B. Give that you are forced to eat spoiled cheese, what is the probability that you selected challege A?. Each year, the cryptids kow oly as tubes idepedetly kill Americas with a average kill rate of 3 Americas per year. (a) Usig a Poisso distributio, approximate the probability that 2 or fewer Americas die ext year from tubes. (b) Briefly explai why a Poisso approximatio here is a reasoable choice ad why oe might use it over the Cetral Limit Theorem (the ormal approximatio)?. Let X ad Y be joitly cotiuous radom variables with joit desity c ( y) < x < y, < y < f X,Y (x, y) = (a) Fid c. Page 2
(b) Give that Y > 2/3, what is the probability that X < /2? 2. Let X be a radom variable with distributio x < F X (x) = 2 x x < 2 2 x (a) Is X cotiuous, discrete, or either? Remember to give me a brief justificatio. (b) Fid E[X]. 3. You roll two fair six-sided dice. Let X be the miimum value of the two rolls showig. For example, if you roll a 3 ad a 4, the X is 3, sice mi(3, 4) = 3. Similarly, if you roll a 5 ad a 5, the X is 5, sice mi(5, 5) = 5. (a) What is a reasoable sample space Ω for this experimet? (b) What is the state space S X of X? (c) Fid the probability mass fuctio p X of X. 4. Carbo-4 has a half-life of approximately 573 years. Orgaic life cotiually repleishes its stock of carbo-4 util death at which poit the carbo-4 decays without repleishig. A orgaism with about 6 carbo-4 particles just died. Let N t cout the umber of decayed carbo-4 particles after a time t of the orgaism s death (where t is measured i years). Based o the iformatio just described, it is reasoable to model (N t ) t as a Poisso process with parameter λ =.725. (a) Assumig that (N t ) t is a Poisso process with parameter λ =.725, what is the probability that two or more carbo-4 particles decay durig oe year followig the death of the orgaism? Please do t leave your aswer as a ifiite sum! (b) Assumig that (N t ) t is a Poisso process with parameter λ =.725, what is the probability that o carbo-4 particles decay durig the first year followig the orgaism s death, ad two carbo-4 particles decay durig the followig two years? 5. Suppose that the momet geeratig fuctio for a radom variable X is give by M X (t) = ( t) 2 (a) Fid a formula for the geeral th momet of X. That is, i terms of, fid E[X ]. (b) Fid Var(X). 6. Suppose that X i } i= are i.i.d. radom variables with mea µ < ad variace σ2 <. Fix some ε >. Use the Cetral Limit Theorem to approximate P ( ε < X i µ < ε ). i= Leave your aswer i terms of the stadard ormal distributio Φ(a) = P (N(, ) a). Your aswer should deped o ε,, ad σ 2. ( ) Hit: X i µ = X i µ. i= i= Page 3
7. Let X be a radom variable with cumulative distributio fuctio s < F X (s) = 2 s s < 2 2 s (a) Let Y be the radom variable defied by Y = l(x) (here, l is the atural log). Fid the cumulative distributio fuctio F Y of Y. (b) Fid E[l(X)]. 8. Let X ad Y be joitly discrete radom variable with probability mass fuctio p X,Y (s, t) described by the followig table X Y 2 2 4 4 8 9 6 6 (a) What is the state space S X of X ad the state space S Y of Y? (b) Calculate E[ X Y ]. (c) Are X ad Y idepedet? Make sure to justify your aswer. 9. The microorgaism E.coli livig i your body has the average lifespa of hours. Right at this momet you, the E.coli whisperer, befried oe of these livig E.coli. (a) If the lifespa of the E.coli is give by a expoetial radom variable, what is the probability that your E.coli fried is still alive i five hours from ow? If you are uable to aswer this questio, make sure to give a good ad brief explaatio why you ca t aswer it ad what more iformatio you would eed to be be able to aswer. (b) If the lifespa of the E.coli is uiformly distributed betwee ad 2 hours (i.e., has a Uif(, 2) distributio), what is the probability that your E.coli fried is still alive i five hours from ow? If you are uable to aswer this questio, make sure to give a good ad brief explaatio why you ca t aswer it ad what more iformatio you would eed to be be able to aswer. 2. A ur iitially cotais 5 red marbles ad 7 blue marbles. Bored out of your mid, you decide to play a game that goes as follows. Durig each roud, you radomly pull a marble out of the ur. If the marble you chose was red, you retur the marble back ito the ur alog with more blue marble. If the marble you chose was blue, you put the marble back ito the ur alog with 2 more red marbles. What is the probability that o the first roud you drew a red marble, ad o the third roud you drew a blue marble? 2. The (X, Y ) locatio of the Mothma lads withi a 2 mile radius, cetered at dowtow Poit Pleasat, West Virgiia (thik of a giat 2 mile radius dartboard cetered at Poit Pleasat Page 4
ad Mothma as a dart ladig at coordiate (X, Y ) o this dartboard). It turs out that X ad Y are joitly cotiuous with joit desity k s 2 + t 2 4 f X,Y (s, t) = where k is some costat. (a) What is k? (This ca be very quick with appropriate justificatio!) (b) What is P (X < Y )? (Agai, this ca be very quick with appropriate justificatio!) (c) Your professor is a sucker. You ivite her to play a game: if Mothma lads withi mile of dowtow Poit Pleasat, the you pay your professor $; if Mothma lads betwee ad 2 miles from dowtow Poit Pleasat, the your professor will pay you $8. Your professor agrees to play. How much moey do you expect to wi? 22. Followig her true dream, your professor opes a surf shop (ad school) i Puerto Rico. Suppose that for each i, we let X i be the umber of customers that make a purchase at your professor s shop o day i. Assume that the collectio X, X 2, X 3,... are idepedet ad d idetically distributed such that X i = Pois(). Use the Cetral Limit Theorem (i.e., a ormal approximatio) to approximate the probability that at most 7 customers made a purchase withi the first 9 days. That is, approximate P ( 9 X i 7 ) usig the Cetral Limit Theorem. You ca leave your aswer i terms of Φ, the CDF of a stadard ormal radom variable. i= Page 5