STA 104 Name:MTH 135 Probability Second Test2:15-3:30 pm Thursday, 20 November 1997You may use a one-sided 8.5 × 11” formula page but may not consult your books, notes,or neighbors. Please show your work for partial credit, and circle your answers. Pointsare awarded for solutions, not answers, so correct answers without justification willnot receive full credit. Please give all numerical answers as fractions in lowest terms oras decimals to four places. When you have finished, please sign the Duke Honor Codepledge.I have neither given nor received aid on this examination.1. /202. /203. /204. /205. /20/100Page 0 of 5STA 104MTH 135 Probability Second Test1. The random variable X has a uniform distribution on the interval [0, 2], while Yhas an exponential distribution with density function f(y) = e−y, for y > 0; X and Yare independent. Be sure to show your work below, as you find:a. E[X] = P[X > 2] =b. E[Y ] = P[Y > 2] =c. P[Y > X] =d. The probability density function for Z =√Y :fZ(z) =Page 1 of 5STA 104MTH 135 Probability Second Test2. Let X and Y have joint density functionf(x, y) =1ye−xy−y, 0 < x < ∞, 0 < y < ∞; f (x, y) = 0, other x, y.a. Find P[Y > 2] =b. Find P[X > Y ] =c. Find P[X > 2|Y = 2] =Page 2 of 5STA 104MTH 135 Probability Second Test3. Two coins fall “Heads up” with probabilities p1and p2, independently, for somenumbers 0 < p1< 1, 0 < p2< 1, and hence fall “Tails up” with probabilities q1= 1 − p1and q2= 1 −p2, respectively. Both coins are tossed.a. What is the probability that they show the same face?b. If they do show the same face, what is the probability that the face they bothshow is Heads?c. If we toss the two coins repeatedly until the first time they show different faces,what is the expected total number of Heads that will appear on the coins, up to(and including) the toss on which they differ?Page 3 of 5STA 104MTH 135 Probability Second Test4. Good news! An anonymous donor has agreed to give every Duke student aThanksgiving Bonus. Bonus amounts (in dollars) will be independent random variables,each with density function f(x) = .01e−.01x, for x > 0. Let X be the amount of yourbonus, let S be the total amount of money the generous Donor must give, and assumefor this problem that Duke has exactly 5,000 students enrolled. There is no need to doANY integration for solving any part of this problem.a. What is the probability distribution of X? Give its name and the value(s) of anyparameter(s). Also give the mean, µX.b. What is the probability distribution of S? Give its name and the value(s) of anyparameter(s). Also give the mean, µS(which you can find even if you don’t knowthe distribution).c. Let Y be the number of students whose bonus is less than ten cents. Give theexact distribution of Y AND pick which (if any) distribution would be a goodapproximation: Poisson, Geometric, normal, or none of the above. If one ofthese is a good approximation, tell why and give the parameter(s); if not, explainwhy.d. Maybe you are exceptionally lucky. Imagine that you talk to all your friends, oneat a time, until you finally find someone with a bigger bonus than yours. Whatis the probability that you must ask k (or more) friends before finding one witha bigger bonus than yours? I.E., what is the probability that the number N offriends you must ask to find one with a bigger bonus than yours satisfies N ≥ k?Note this question concerns the marginal distribution of N, not the conditionaldistribution given X.P[N ≥ k] =Page 4 of 5STA 104 Name:MTH 135 Probability Second Test5. A new drug is being tested in a trial with 100 subjects, to discover the probabil-ity p that the drug is effective in reducing blood pressure. Of course there may be un-expected side-effects; let θ be the probability that a subject will experience an adversedrug reaction of some kind. The subjects are treated one at a time; if ANY ONE has anadverse reaction, the trial will be halted. Let S be the total number of subjects treated(S = 100 if there are no adverse reactions, but S = s < 100 if subject s has the firstadverse reaction), and let X be the number of subjects who respond favorably to theblood-pressure treatment. Adverse reactions are independent of effectiveness—it is pos-sible for the treatment to work for a subject who nevertheless experiences an adversereaction.a. What is the probability that the trial is not halted early?P[S = 100] =b. What is the conditional probability of x subjects responding successfully, if thereare no adverse reactions?P[X = x|S = 100] =c. Are X and S independent? Why?d. Give an expression for the marginal probability mass function for X; you need notevaluate the resulting sum (Hint: think about the distribution of S).Page 5 of 5