Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions

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A university committee consists of six faculty members and four students.If a subcommittee of four persons is to be chosen randomly from the com¬mittee, what is the probability that it will consist of at least two faculty members? Compute this probability from the hypergeometric distribution and
If twelve cards are to be drawn at random without replacement from a stand¬ard 52-card deck, what is the probability that there will be exactly three cards of each suit?
One of the key assumptions in the derivation of the binomial distribution is that the various trials are independent. Is this same assumption made in the calculation of probabilities for poker hands or in deriving the hypergeometric rule? Explain.
An instructor in an elementary laboratory class in psychology knows that out of 50 students, 12 are freshmen, 25 are sophomores, 11 are juniors, and only 2 are seniors. He assigns these students to particular laboratory sections at random, ten to a section. What is the probability that a particular
Suppose that cars arrive at a toll booth according to a Poisson process with intensity 5 per minute. What is the probability that no cars will arrive in a particular minute? What is the probability that more than 8 cars will arrive in a particular minute? What is the probability that exactly 9 cars
Suppose that incoming telephone calls behave according to a Poisson process with intensity 12 per hour. What is the probability that more than 15 calls will occur in any given one hour period? If the person receiving the calls takes a fifteen minute coffee break, what is the probability that no
Consider the occurrence of misprints in a book, and suppose that they occur at the rate of 2 per page. Assuming a Poisson process, what is the expected number of pages until the first misprint? Wbat is the probability that the first misprint will not occur in the first page? Comment on the
Prove that the variance of the Poisson distribution is equal to X.
Think of two applications in your area of interest which involve the Bernoulli process and two which involve the Poisson process.
Accidents occur on a particular stretch of highway at the rate of 3 per week.What is the probability that there will be no accidents in a given week?What is the probability that there will be no accidents for two consecutive weeks? Compute these probabilities (a) using the Poisson distribution,(b)
Comment on the following statement: “The exponential distribution is to the Poisson distribution as the geometric distribution is to the binomial distribution.” In general, what is the difference between a Bernoulli process and a Poisson process?
If the median of an exponential distribution is 2, find X.
If the first quartile of an exponential distribution is 2, find X.
Suppose that the instructor in your History of Babylonia course informs you that on the final examination two students received grades of 60 and 30 respectively, and that the standardized scores corresponding to these grades are 1 and —1, respectively.(a) What is the mean and the standard
In this problem, and those to follow, the notation N (n,
If X is distributed as N (3, 4), use the table to find:(a) P(l< X< 7), P(X< 5), P(-2< X< 1.5)(b) a number a such that P ((X — 2)
If the .35 fractile of a normal distribution is 105 and the .85 fractile is 120, find the mean and standard deviation of the distribution.
Let us assume that at a certain university the actual time that a graduate student spends working toward his Ph.D. degree is approximately normally distributed with a mean of 200 weeks and a standard deviation of 50 weeks.Given 1000 students beginning graduate work toward the Ph.D. at the same
The mean score of 1000 students on a national aptitude test is 100 and the standard deviation is
We will assume that the distribution of such scores is well-approximated by a normal distribution. Find how many students have scores:(a) between 120 and 155 inclusive(b) greater than 95(c) less than or equal to 175(d) 5 or more points below the score cutting off the lowest 25 percent(e) scores 10
A sample of four independent observations is taken from a normal population with mean y and variance a2.(a) What is the probability that all four sample observations will fall between the limits y -f- a and y— a?(b) What is the probability that all four sample values will fall between the limits
Why is the normal distribution of such importance in statistical work? Try to think of some applications in your area of interest which might involve the normal distribution.
Under what conditions would the continuous uniform distribution be appli¬cable? It is also possible to consider a discrete uniform distribution in which all of the possible values of the random variable have the same probability.How might this distribution relate to the discussion in Section 2.6?
If the random variable is uniformly distributed on the interval from a to b with mean 6 and variance 3, find a and b.
Find the mean and variance of the beta distribution with parameters r = 2 and N = 6, and graph the density function. Do the same for the following beta distributions:(a) r = 4, N = 6 (b) r = 4, N = 12 (c) r = 8, N = 12 Explain how the different values of r and N affect the shape and location of
If the mean and variance of a beta distribution are 2/3 and 1/72, respec¬tively, find r and N.
What is required to determine whether a given object is or is not a member of a given set?
What is the role of the universal set in set theory? What is the role of the empty set?
Specify the following sets by first listing all their members and then stating a formal rule:(a) The set of all positive integers less than or equal to 12.(b) The set of all positive even integers less than or equal to 12.(c) The set of all odd positive integers less than or equal to 12.(d) What is
Let the set A = {0, 1, 2, 3, 4, 5}.List the elements in each of the following sets that are also in the set A.(a) {x | x + 3 = 5}(b) \x | z2 + 2x + 1 = 0}(c) {x\ 2x - 3 = 7)(d) {x| z2> 1}(e) {*| x + 2 > 6}Let A = {-2,-1,0, 1,2}List the elements of A that are also in each of the following sets:(a)
Let the universal set be the set of real numbers, and let A — {x\x2>l},B = {x\x2 + 2x + 1 = 0},andC = {x | 2x - 3 < 7}.Specify the following sets:(a) A U B (d) 4nC(b) AUBUC (e) (4uB)nC.(c) C — A
Consider the following sets:W — Set of all students at the University of Michigan.A = Set of all males.B = Set of all graduate students.C = Set of all students who take mathematics courses.D = Set of all students who take business courses.Describe in words the following sets:(a) AU(BUC) (d)
A certain journal in statistics makes references to various other journals and also to itself. Suppose that in a sample of 100 articles from this journal the numbers of times the journal referred to journals A, B, and C (itself) is as given below:A 60 B 40 C 70 A and B 32 A and C 45 B and C 38 A,
Make Venn diagrams to verify that the following theorems on sets hold. Use some systematic shading or coloring to clearly distinguish the relevant sets.(a) A n (B U C) = (A fl B) U (A fl C)(b) (Aflfi)C 1C (A U B)(c) A - BC A(d) if A C B and C C D, then (A U C) C (B U D)(e) if AD Rand CD D, then
If A = {0, 1, 2, 3}, list all possible subsets of A.
What distinguishes a Cartesian product from any other set?
When is a relation a function?
Let A = {2, 3, 4, 8, 9}. Graph these mathematical relations, each of which are subsets of A X A:(a) {(2,9), (4,9), (8,9), (9,9)}(b) {(2,3), (2,4), (2,8), (2, 9)}(c) {(2,4), (3,9)}(d) { (2, 3), (3, 4), (4, 8), (8, 9)}(e) {(3, 2), (4,3), (8,4), (9,8)}(f) Which of these relations are functions?
List all the ordered pairs that can be obtained from the set X = {1, 2, 3, 4, 5), corresponding to each of the following relations:(a) Is a multiple of. (d) Is equal to.(b) Is greater than. (e) Is greater by two.(c) Is less than. (f) Is less by two.
Suppose four people are sitting at a table. Let U be the set of people sitting at a table, U = {A, B, C, D]. The people are seated as in the figure.A D B CDraw the graphs of the following relations that are subsets of U X U and specify which of the relations are functions.(a) { (x, y) | x sits next
In each of the following, list the elements of the relation and give its domain and range, where U — {1, 2, 3, 4, 5}, and (x, y) is a member of U X U.(a) \{x,y)\x = 2} (d) {(x, y) \x = (1/2)?/}(b) {(x,y)\y = 5\ (e) {{x,y)\y
Consider the following sets of letters: {a,b, c,d,e} = A,{u, v, w, x, y, z) = B, C — {a,e, i, o, u}. Let W = A U B U C.List the members of:(a) Afl C (e) A - C(b) Au(BnC) (f) AXC(c) B\jC (g) AX A.(d) (AnB)u (Bn C)
Given A = {x | x2 < 16}, what is the Cartesian product A X A? Draw the graphs of the following relations on A X A and specify which of the relations are functions.(a) {(x, y) | x + y = 3} (d) {(x, y) \ y3 = 64 — x}(b) } (x, y) | x2 + y2 = 4} (e) {(*, y) | y > x + 3}.(c) { (*, y) | z2 = 16 — y]
In Exercise 19, find the domains and the ranges of the relations in (a), (b),(c), (d), and (e).
For each statement, tell whether it is true (for any sets A, B, and C) or false, and if it is false, draw a Venn diagram to demonstrate that it is false.(a) (A — C) U {B- C) = {A US) - C
Prove, by set algebra, that(a) (AUB)= (A n5) (b)_ (TrTB) = (AUJ5)/or (a): Show that (A U i?) U (A ClB) = IP and that (AU5)fl(A fl B) = 0, so that the set (A U B) must be the complement of (A fl 5).]
Given that A, B, and C are not mutually exclusive, express their union A U B U C in terms of the union of three sets which are mutually exclusive.
Explain, briefly, the connection between the theory of sets and probability theory.
A black die (B) and a white die (TE) are each tossed. Let 6 indicate the value of the B die, and let w indicate the value of the W die. (That is, let the number of spots coming up on a die be its numerical value for that toss). Specify the sample space completely, and then find the probabilities
A letter is selected at random from the English alphabet. What is the prob¬ability that:(a) the letter is a vowel(b) the letter is a consonant(c) the letter occurs in the last ten positions of the alphabet, in the con¬ventional ordering?
Two distinguishable coins are tossed two times each. Find the probability for each of the following events, given that the coins are fair. Start by specify¬ing the elementary events making up the sample space.(a) at least two heads occur(b) at most two heads occur(c) the number of heads for coin 1
Why is it that almost all elementary discussions of, and computations in¬volving, probability rely on the assumption of equally probable elementary events? Is there anything about the theory of probability itself that makes this assumption necessary? What if elementary events cannot be assumed
A fair coin is tossed N times, and each time either a head or a tail occurs.If the elementary events for this experiment are viewed as the possible results of N tosses, how many different elementary events are there?
A young man has a little black book that contains the names and phone numbers of girls who are potential dates. Suppose that he has the names of 3 redheads, 18 brunettes, and 6 blondes in his book, and that the book has a total of 30 pages. He decides to select one page at random from the book in
Show by a Venn diagram that(a) P(ADB) = P(A) - P(Al)B)(b) if BC A, then P(A - B) = P(A) - P(B)(c) P(A n B) + P(A n C) = P[A n(BUC)] + P(4flBn C).
Let A\, A2, and A3 be events defined on some specific sample space S. Find expressions involving only union, intersection, and complementation for the probability that:(a) exactly two of Ai, A2, A3 occur simultaneously(b) at least two of Ax, A2, A3 occur simultaneously(c) at most two of the events
Compute the probabilities for (a)-(c) in the preceding exercise if:(a) P(A1) = 1/2, P(A2) = P(A3) = 1/4 P(^i0 A2) = P(Axn A3) = P(A2fl A3) = 1/8 P(Aifl A2n A3) = 1/32;(b) P(A1) = 1/3, P(A2) = 2/3 p (A! n As) = p (Ai n a3) - p (a2 n a3) = p (ax n a2 n a3) = o.
A card is drawn from a well-shuffled standard playing deck. What is the probability that it is:(a) either a spade or the queen of hearts(b) an even numbered card (not counting face cards) but not a spade or a heart(c) a spade which is not the ace or king, or a heart which is not the queen or
A famous problem in probability theory is the following: if an “ace” is the occurrence of the number “one” on the toss of a die, find the probabilities of(a) obtaining at least one ace in four tosses of a fair die(b) obtaining at least one “double ace” in twenty-four tosses of a pair of
Find a general formula, similar to Equation (2.5.7), for the probability of the union of three events, P (A U B U C). [Hint: See Exercise 23, Chapter 1.]
Suppose that a fair die is thrown until a six appears. Find the probability that this will take more than three throws.
Prove that P(A fl B) < P(A) < P(A U P) for any events A and B.
In one year, the Parliament of a European country included 45 members of the Liberal party, 38 members of the Conservative Party, and 15 members of the Labour Party. The Prime Minister wished to appoint a commission of 10 persons from Parliament, consisting of 5 Liberals, 3 Conservatives, and 2
If there are 5 people in a room, what is the probability that no two of them have the same birthday? In general, if there are r people in a room, what is the probability that no two of them have the same birthday? What assump¬tions did you make in computing these probabilities?
If we draw five cards from a well-shuffled standard deck of 52 cards, what is the probability of obtaining(a) four of a kind(b) a royal flush (Ace, King, Queen, Jack, and Ten from the same suit).
In how many distinguishable ways can the letters in the word “sassafras”be ordered?
Prove Pascal’s rule:O-C:>(";')
Prove that
How many ways can a starting team of 11 players be chosen from a squad of 30 football players, ignoring the position to be played by each player? If the squad consists of 10 backs and 20 linemen, in how many ways can a starting team of 4 backs and 7 linemen be chosen?
State how many ways can 8 persons be seated in a row of 8 seats if(a) there are no restrictions on the seating arrangement(b) person A must sit next to person B, but there are no restrictions on the remaining six persons(c) there are four men and four women, and no two men (or two women)can sit
How many ways can 8 persons be seated around a circular table, taking into account only the relative location of the 8 persons (that is, if all 8 persons get up and move one chair to the left, the new arrangement is considered to be indistinguishable from the old arrangement)?
Suppose that you want to arrange 3 statistics books, 2 mathematics books, and 4 novels on a bookshelf. Calculate how many arrangements are possible if(a) the books can be arranged in any manner(b) the statistics books must be together, the mathematics books must be together, and the novels must be
The so-called “gambler’s fallacy” goes something like this: in a dice game, for example, a seven has not turned up in quite a few rolls of a pair of honest dice; now a seven is said to be “due” to come up. Why is this a fallacy?
Bernoulli’s theorem has occasionally been called “the link between the mathematical concept of probability and the real world about us.” Comment on this proposition.
Take a fair die and throw it 300 times, recording the number which appears at each throw. Find the frequency of each of the six possible events after 10, 20, 50, 100, and 300 throws, and comment on these results in light of Bernoulli’s theorem.
Find the probability of event A if the odds in favor of A are:(a) 2 to 1 (b) 1 to 2 (c) 3 to 7.
Find the odds in favor of event A if the probability of event A is(a) .50 (b) .20 (c) .875.
Explain why the subjective interpretation of probability can be thought of as an extension of the long-run frequency interpretation. Are there any restrictions on the types of events for which subjective probabilities can be used?
(a) What is your subjective probability that it will rain tomorrow?(b) What in your opinion are the odds in favor of rain tomorrow?(c) Are your answers to (a) and (b) consistent? If not, why not?(d) When “precipitation probabilities” are given out to the public, how do you interpret them? How
Three football teams are fighting for the league championship. A sportswriter claims that the odds are even, or 1 to 1, that Team A will win the champion¬ship, the odds are even that Team B will win the championship, and the odds are even that Team C will win the championship. Is this reasonable?
In the preceding problem, the sportswriter states that he is willing to bet for or against Team A at even odds, and likewise for Teams B and C. Even without knowing anything about the teams, would you like to bet with him?Explain.
In what sense is it true that a joint event emerging from a compound experi¬ment can always be regarded as an elementary event?
As we saw in Chapter 1 of the text, a relation is a subset of a Cartesian prod¬uct, such as A X B. In a statistical relation, what would be the elements of the sets A and B making up the Cartesian product?
Consider an experiment involving a Red and a Black die. One of the two dice is selected at random, and then the selected die is rolled.(a) Let C = {R, B} and G = {1, 2, 3, 4, 5, 6}. Find C X G = S, the sample space for the above experiment.(b) Display this sample space in graphic form.(c) Find the
Suppose that you have three dice, Red, Black, and Yellow. Now one of the dice is selected and rolled.(a) Find the sample space S for this experiment.(b) Find the elementary events associated with each of the following events, as well as the probabilities of these events:(i) the die selected is
Let A be the event “a person is a male,” and B the event “a person is blue¬eyed” when individuals are chosen at random from some well-defined group of persons. Also let P(A) = .60 P(B) = .20 P(A | B) = .10.Find: (a) P(AUP) (c) P(A U B), P(A fl B)(b) P(A UP), P(A UP) (d) P{(lnB)U(inB)j.
If P(A) = .6, P(B) = .15, and P(B\ A) = .25, see if you can complete the following table:B B AA Also, find the following probabilities:P(B\A) P(AUB)P(A | B) P{(AnB)U (AnB)}.
Consider some sample space, in which events Ah A2) and A3 are defined.You are given the following probabilities:(a) P(Ai) = 1/2, P(A2) = P(A-i) = 1/4 P(Axn A2) = P(Axn A3) = P(A2n As) = 1/8 P(Axn A2fl As) = 1/32.Find P(Ax|A2), P^nA^As), P^A^As), P(Ai U A21 A3).
In Exercise 41, imagine that P(Ai) = 1/3, P(A2) = 2/3, and that P(A3) = 0. Also that P(Ax fl A2) = P(Ai fl A3) = P(42n43) = 0. Are Au A2, and A3 independent events here?
Explain the difference between mutually exclusive events and independent events. Is it possible for two events A and B to be both mutually exclusive and independent? Explain.
If P{A) = 0, does it make any sense to consider the conditional probability P(B | A) ? Explain.P(A)
Show that if A fl B = 0, then P(A I A U B)=-.1 P(A) + P(B)
Suppose that a survey of 200 people at a certain college town has provided the following information about a proposal to allow more liberalized sale of beer:
and34.4.Themodeforeachgenderwas0.WouldaPoissonGLMwithanindicator variableforgenderbesuitableforthesedata?Whyorwhynot?
AquestioninaGSSaskedsubjectshowmanytimestheyhadsexualintercourseinthepreceding month.Thesamplemeanswere5.9formalesand4.3forfemales;thesamplevarianceswere
Refertothepreviousexample.Usingallfourexplanatoryvariablesinthedatafileaspotential predictors, andconsideringcolorandspineconditionasquantitative,selectamodelusingAIC or BIC.Interpretitseffects.
RefertothePoissonloglinearmodelin Section 7.4.2 for thehorseshoecrabs,withweightand color asexplanatoryvariables.(a) Whenyouviewascatterplot,identifyanunusualobservation.Re-fitthePoissonGLM without itanddescribehowtheestimateoftheweighteffectchanges.(b)
Inthe Crabs data fileintroducedin Section 7.4.2, thevariable y indicates whetherafemale horseshoecrabhasatleastonesatellite(1 = yes,0 = no).(a) Fitamain-effectslogisticmodelusingweightandcategoricalcolorasexplanatoryvari-ables. Conductasignificancetestforthecoloreffect,andconstructa95%confidence
Inatennismatchwithplayers a andb, let Πab betheprobabilitythat a defeatsb. The Bradley–Terrymodel is log(Πab~Πba) = βa − βb, with anidentifiabilityconstraint,suchas βc = 0 for playerc. For T players,thismodels T(T −1)~2 probabilities by (T − 1) parameters. Fitthemodeltothe Tennis data
for the priordistributions.(b) ToconductaBayesiananalysiswithuninformativepriors,use σ = 100 in theBayesian analysis, verifyingresultssuchasshownin Table7.4. Comparewhatyoulearnaboutthe effect ofNVwiththispriorandwiththehighlyinformativepriorhaving σ = 1.

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Foundations Of Statistics For Data Scientists With R And Python 1st Edition Alan Agresti - Solutions

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