Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

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3.3.16 Find the number ofpolygonal lines (as in Problem 3.3.15) that join the points(2,3) and (16,5) and: (i) Never touch the z-axis. (ii) Never touch the line y = 7.
3.3.15 Find the number ofpolygonal lines with vertices (z, C(z)), where C(z) is as inExample 3.11 andwithpossibleedges leading from (5, C(z)) to ( z f l , C(z)+l)or (z + 1, C(z) - l), connecting the points: (i) (0,O) and (10,O). (ii) (0,O) and(10,5). (iii) (3, -2) and (8,l).
3.3.14 Compute probabilities P ( z ) of winning z numbers in lotteries, where the player chooses: (i) 5 out of 44 numbers. (ii) 6 out of 55 numbers.
3.3.13 Suppose that n balls are distributed in k boxes (n 2 k ) . Find the probability that there are no empty boxes. [Hint: Let A(n, k ) be the number of allocations of balls such that no box is empty. Show that A(n,k ) = ( i )A( n- i, k - 1).Use boundary condition A(n, 1) = 1 to find A(n, k).]
3.3.12 (i) If n balls are put at random into n boxes, what is the probability of exactly one box remaining empty? (ii) Suppose that we distribute n balls in k boxes (n 2 k ) , labeled 1, . . . , k . Show that the number of different ways this can be done, so that there is at least one ball in each
3.3.11 A poker player has three-of-a-kind. He discards the two unmatched cards, and obtains two new cards. Find the probability that he will have: (i) Three-of-akind.(ii) Four-of-a-kind. (iii) A full house.
3.3.10 A poker player has 3 0 , 7 ~ , 8 0 , 9 0Q,& She discards 3 0 and QC and obtains 2 cards. What is the probability that she will have: (i) A straight flush. (ii)A flush, but not a straight flush. (iii) A straight, but not a straight flush.
3.3.9 A poker player has 3 0 , 7 0 ,8 4, 9 0 ,Q He discarids 3.0 a nd Qa and obtains 2 cards.' (i) What is the probability that he will have a straight? (ii) Answer the same question if Q 1 i s replaced by 5 1 ( i.e., he discards 30 and 51).
3.3.8 Find the probability that a poker hand will contain two pairs, one red and the other black.
3.3.7 (Poker Hands) Find the probability of each of the following hands:(i) Royal flush (ace, king, queen, jack, and 10 in one suit),(ii) Straight flush (five cards of one suit in a sequence, but not a royal flush),(iii) Flush (five cards in one suit, but not a straight flush nor a royal
3.3.6 Peter lives at the corner of 2nd Avenue and 72nd Street. His office is in the building at a corner of 7th Avenue and 78th Street. The streets and avenues in the city form a perpendicular grid, with no streets or passages in the middle of the blocks.Peter walks to work along either street or
3.3.5 How many ways can one order the deck of 52 cards so that all four kings are next to each other?
3.3.4 What is the probability that a randomly chosen committee of 50 senators contains one senator from each of the 50 states?
3.3.3 A committee of size k is to be formed out of the US Senate. How large must k be in order for the probability of at least one senator from Ohio being included to exceed the probability of no senator from Ohio being included?
3.3.2 A committee of size 12 is to be formed out of the US Senate. Find the probability that: (i) Both senators from Ohio will be represented. (ii) Exactly one senator from Ohio will be represented. (iii) No senator from Ohio will be represented.
3.3.1 Whichislarger: (i)('::') or ('g:'). (ii) (y!:) or (y!:). (iii) ('g:') or (';:I).
3.2.20 Assume that in a corporation the number of votes of a shareholder equals the number of his shares. If a simple majority is required, determine the voting powers in a corporation in which there are three shareholders, A with 500,000 shares, B with 499,999 shares, and C with 1 share.
3.2.19 The United Nations Security Council consists of five permanent members(China, France, the Russian Federation, the United Kingdom, and the United States)and 10 nonpermanent members. For an issue to pass, it must receive a unanimous vote of all five permanent members (each has veto power) and
3.2.18 Find the voting powers in a voting body of 2 n members, among them the chairperson. Everyone (including the chairperson) has one vote, and the majority carries the issue. In case of a tie, the chair’s vote prevails.
3.2.17 (i) Determine the voting power in a committee of five persons, each having one vote, with a simple majority needed to pass the issue, under the additional condition that A has veto power (i.e., positive vote of A is necessary to pass the issue).(ii) Assume, in addition, that B has two votes.
3.2.16 Determine the voting powers in a committee consisting of six persons, with A having three votes, B having two votes, all others having one vote, and again, a simple majority being required to pass the issue.
3.2.15 Generalize the situationofProblem 3.2.14 assuming that A has k votes (other conditions being the same). What is the smallest k for which A is a dictator (i.e., every other member has zero power)?
3.2.14 Find the power of A in a committee of n persons with A having two votes and every other member having one vote. Assume that simple majority is needed to carry the issue.
3.2.13 Find the number of three-digit integers (i.e., integers between 100 and 999)that have all digits distinct. How many of them are odd?Problems 3.2.14 through 3.2.20 concern the concept of voting power, as introduced by Shapley and Shubik (19541, Alhmgh this concept is not direcllyrelated to
3.2.12 Lete, be the probability that exactly two people in a group of T have the same birthday, and let p , be the probability that everybody in the group has a different birthday. (i) Show that r ( r - 1)e, = ~ 730 Pr-1.(ii) Use the multiplication rule (Theorem 3.2.2) to show that r ( -~ 1 )2 365
3.2.11 Express the product of odd integers 1 x 3 x . . x (272 + 1) using factorials.This product is sometimes denoted (2n + l)!!. [Hint: Start with an easier task of using factorials to express the product of even integers 2 x 4 x . . x (2n).]
3.2.10 A restaurant menu has five appetizers, three soups, 15 entrees, and three desserts. (i) Assuming you are going to order one item from each group, how many possible dinners can you order? (ii) Assume you come to the restaurant with a friend who is as hungry as you are. How many different
3.2.9 Susan has five dresses, three skirts, four blouses, three pairs of shoes, and two hats. She always wears shoes, and either a dress or a blouse and a skirt. She may or may not wear a hat. (i) How many different combinations can she wear? (ii)Suppose Susan can afford buying either a dress or a
3.2.8 12 girls and 17 boys go to a dance. (i) How many possible dancing pairs (boygirl)may be formed? (ii) The dance floor can accommodate at most 1 1 pairs at a time. If each dance lasts 10 minutes and is followed by a 2 minute break, how much time, at least, will elapse before each boy will have
3.2.7 Five men and five women are to be seated in a row of ten chairs. Find the number of possible arrangements if: (i) The men are required to sit in alternating seats. (ii) No two men are to be seated next to each other.
3.2.6 Seated at random at a round table with n seats are n people, among them John and Mary. (i) Answer questions (iHiv) of Problem 3.2.5. Anything peculiar about the answer to (iii)? (ii) Assume that n = 2k. Find the probability that John and Mary sit facing each other ( e g , numbers 1 and 7 on
3.2.5 Seated at random in a row of n seats are n people, among them John and Mary.Find the probability that: (i) John sits next to Mary. (ii) John sits next to Mary on her right. (iii) John sits somewhere to the right of Mary. (iv) John and Mary sit exactly two seats apart.
3.2.4 Determine the number of 0’s at the end of 16! and 27!.
3.2.3 A two letter code is to be formed by selecting (without replacement) the letters from a given word. Find the number of possible codes if the word is: (i) CHART.(ii) ALOHA. (iii) STREET.
3.2.2 A skyscraper is 40 stories tall. Five people enter the elevator on the first floor.Assuming each person is equally likely to get off at any ofthe 39 floors 2 , 3 , . . . ,40, what is the probability that all people will get off at different floors? Find the exact value, and then derive and
3.2.1 A certain set contains n distinct elements. Find n if the number of: (i) All possible permutations of length 2 equals 90. (ii) Permutations of length 3 is 10 times larger than the number of permutations of length 2.
2.7.3 Suppose that Tom is confronted with the choice between two options: 01, which is simply to receive $1,000,000, or 0 2 , which is to receive $5,000,000 with probability 0.1, receive $1,000,000 with probability 0.89, and receive $0 with the remaining probability 0.01, After some deliberation
2.7.2 Refer to Problem 2.7.1. Tom would agree on the following conditions: Peter pays him $50 and tosses a coin, winning the ticket if it comes up heads, and otherwise losing $50. In such situation, should they both agree that Peter buys the ticket from Tom for S 150?
2.7.1 Peter and Tom attend the same college. One day Tom buys a ticket for a rock concert. Tickets are already sold out and are in great demand. Peter, who does not have a ticket, agrees to play the following game with Tom. For a fee of $25, Peter will toss a coin three times, and receive the
2.5.4 A number X is chosen at random from the series 4,9, 14, 19, . . . , and another number Y is chosen from the series 1, 5,9, 13, . . . . Each series has 100 terms. Find P ( X = Y).
2.5.3 Use formula (2.7) to find the number of primes not exceeding 100. [Hint:Assume that you sample one of the numbers 1,2, . . , , 100. Let Ai be the event “the number sampled is divisible by a,” Determine p = P(A2 U A3 U A5 U AT). Then the answer to the problem is lOO(1 - p ) + 3 (why?).]
2.5.2 A die is tossed three times, with outcomes X1 , X2, and X3. Assuming that all 216 possible outcomes (z1,22,53)a re equally likely, find following probabilities:(i) P(X1 > X2 = X 3 ) . (ii) P(X1 < X2 < X3). (iii) P[max(Xl, X2, X 3 ) = 31.(iv) P[min(X1X, p, X3 ) = 21.
2.5.1 A coin is tossed 7 times. Assume that each of the 27 = 128 possible outcomes(sequences like HTTHHTH of length 7) is equally likely. Relate each outcome to a binary number by replacing H by 1 and T by 0, for example, THHHTTH is 0 1 1 1001= 57. Find the probability that a number generated in
2.4.13 A die is loaded in such a way that the probability of j dots on the top face is proportional to j, for j = 1, 2, . . . , 6. What is the probability that in one roll of the die an odd number of dots will turn up?
2.4.12 A regular die and a die with 2, 3, 5 , 6, 7, and 8 dots are tossed together, and the total number of dots is noted. What is the probability that the sum is greater than or equal to lo?
2.4.1 1 A survey is conducted in a certain city to determine the number of households having electric appliances. It is found that 67% have washing machines (W), 52%have microwave ovens (M), and 48% have dishwashers (D). Furthermore, 32% have(WM), 30% have (MD), 30% have (WD), and 20% have all
2.4.10 A certain public phone is such that it returns the coin with probability a, connects you with the number you dial with probabilityb, and it gives you the connection for free with probabilityc. Let us agree to say that the phone is individually honest if it takes your money if and only if it
2.4.9 A faulty public phone is such that it returns the coin with probability 60%, it gives you the number you dial with probability 20%, and it takes your coin and does not give you the required connection with probability 30%. Find the probability that you will talk with the number you dial for
2.4.8 John and Mary have to take a certain statistics course. John attends 40% of the classes, while Mary misses 20% of the classes. They are both present in class only 32% of the time. Find the probability that: (i) Only one of them is present in class. (ii) They are both absent.
2.4.7 Four events A, B, C, and D are such that A and B are disjoint and D is contained in ( A U B U C)". Moreover, B n C is twice as likely as A n C, while P(AnCc)= P(BnCc)= 0.2,P(D)= P[(AUBUCUD)"=] P(CnACnBC=)0.1. Find the probability that out of the events A, B, C, D: (i) None will occur.
2.4.6 Let A, B, C, D , and E be five events such that P(A) = P(B) = . ' ' =P ( E ) = k and P(A n B ) = P(A n C ) = ' . . = P(D f l E) = p . Moreover at least one of the events A, . . . , E must occur, and the intersection of any three events among A, B, . . . , E is empty. (i) Find p if k = 0.3.
2.4.5 Three events A, B, and C are such that C is contained in the intersection A" n B, P(A) = 0.6, P(A n B) = 0.3, and P(C) = P(AC n B") = 0.1. Find:(i) P(B). (ii) The probability that exactly two of the events A, B, C will occur. (iii)The probability that exactly one of the events A, B, C will
2.4.4 To make the formula P[ (ACBn" )"]= 2 - P(AC-) P(BC-) x valid, x must equal: (1) P(A U B). ( 2 ) P(A n B). ( 3 ) P(AC U B"). (4) None ofthe above.
2.4.3 Find P ( A U B") for events A and B such that P(A) = P(B) = 1/2, and P(AC n B") = 1/3.
2.4.2 Events A, B, and C are such that A n B = A n C = 0 while B n C is not empty. Determine P(A U B U C ) as a function of z if P(A) = P(B) = P(C) =3P(B nc) = 5 .
2.4.1 Let events A, B, and C be such that A contains B and is disjoint from C.Moreover, A is twice as likely as B, three times as likely as C, and half as likely as its complement A". Find P(B U C).
2.3.4 Show that first and third, as well as second and third, schemes of sampling chords (see Bertrand’s paradox) are not equivalent.
2.3.3 (i) A point (a,b ) is selected at random from the square [-ll1 12. Find the probability that the equation ax2 + b s + 1 = 0 has two distinct real solutions. (ii)Answer the same question if the point (a,b) is selected at random from the rectangle A1 5 a 5 A2, B1 5 b 5 B2. Discuss all possible
2.3.2 A bathroom floor is covered by square tiles with side lengtha. You drop a coin with diameterb, where b
2.3.1 Label all statements below as true or false.(i) If A is more likely to occur than A", then P ( A ) > 0.5.(ii) If A occurs whenever B does, then P ( B ) 5 P ( A ) .(iii) If P(A)5 P(B) ,th en whenever B occurs, A does also.(iv) If P ( A ) = 0.75, then A must occur three times out of every
1.4.6 Let S = (-co, +m). Show that the class of all finite unions of intervals of the form [a, b ] , (a, b ) , [a, b), and (a, b], with possibly infinite a or b (intervals of the form [a, co); etc.) forms a field.
1.4.5 Let S = { 0 , 1 , 2 , . . .} be the set of all integers. For A c S, let f n ( A ) be the number of elements in the intersection A n { 0 , l , . . . n}. Let A be the class of all sets A for which the limit q (A) = lim -fn(A)exists. Show that A is not a field. [Hint:L et A1 = { 1,3,5,. . .} and
1.4.4 For 0 < Q < 1, let I ( a ) = {z : 1 - a < z < 1 + a}. Consider a sequence~ 1 ~ , 2. ., . o fnumbers satisfying 0
1.4.3 Let A l , . . . , A , be distinct subsets of S. (i) Find the maximum number of sets (including S and 0) of the smallest field containing A l l . . . A,. (ii) Find the maximum number of sets in this field if A,-1 c A,. (iii) Answer (ii) if A1 c A2 c. . c A,. (iv) Answer (ii) if A1 = . . . = A
1.4.2 Assume that John will live forever. He plays a certain game each day. Let Ai be the event that he wins the game on the ith day.(i) Let B be the event that John will win every game starting on January 1, 2015.Label the following statements as true or false: (a) B = liminf A,. (b) B C liminf
1.4.1 Let B1, B2, . . . be a countable partition of S; that is, Bin Bj = 0 for all i # j, and U iB i= S. Let A, = B, U B,+1 U . . . Find lim A,.
1.3.8 Let Anc be defined inductively by Aoc = A , AnC and AmC U AnC for m, n > 0.
1.3.7 Five burglars, A , B , C, D , and E, divide the loot, consisting of 5 identical gold bars and 4 identical diamonds. Let A j k be the event that A got at least j gold bars and at most k diamonds. Let Bjkl c j k denote analogous events for burglars B, C (e.g., B21 is the event that B got 2, 3,
1.3.6 A standard deck of cards is dealt among players N , S, E, and W. Let Nk, k =1 , 2 , 3 , 4 be the event "N has at least k aces," and let S k , Ek and wk be defined similarly. For each of the events below, determine the number of aces that N has.(i) NI n S1 n El n Wl. (ii) EZ n (WZ u SZ). (iii)
1.3.5 Let A, B, and C be three events. Match, where possible, events D1 through Dlo with events El through Ell. Matching means that the events are exactly the same; that is, if one occurs, so must the other and conversely (see the Definition 1.3.2). (Hint: Draw a Venn diagram for each event D1,. .
1.3.4 In a group of 1000 students of a certain college, 60 take French, 417 take calculus, and 509 take statistics. Moreover, 20 take French and calculus, 17 take French and statistics, and 147 take statistics and calculus. However, 196 students do not take any of these three subjects. Determine
1.3.3 Find X i f (i) A t X = 0. (ii) A+X = A. (iii) A+X = S. (iv) A t X = B.
1.3.2 In the statements below A , B, C and D , are events. Find those statements or formulas that are true.(i) If A n B = A n C then B = C.(ii) A U ( An B)U ( An BC)= A.(iii) A u ( A n €3) U ( A n B C)= B.(iv) If A \ B = C, then A = B u C.(v) ( A u B ) n (C u D ) = ( A n C) u ( A n D ) U ( B n C
1.3.1 Answer true or false. Justify your answer.(i) If A and B are distinct events (i,e., A # B) such that A and BC are disjoint, then AC and B are also disjoint.(ii) If A and B are disjoint, then AC and BC are also disjoint.(iii) If A and B are disjoint, and also B and C are disjoint, then A and C
1.2.9 Referring to Example 1.9, suppose that we modify it as follows: The respondent tosses a green die (with the outcome unknown to the interviewer). If the outcome is odd, he responds to the Q-question; otherwise, he responds to the question“Were you born in April?” Again, the interviewer
1.2.8 Let E be the experiment consisting of tossing a die twice. Let S be the sample space with sample points (2, j ) , i, j = 1 , 2 , . . .6, with i and j being the numbers of dots that appear in the first and second toss, respectively.(i) Let S’ be the sample space for the experiment E
1.2.7 Let & be the experiment consisting of tossing a coin three times, with H and T standing for heads and tails, respectively.(i) The following set of outcomes is an incomplete list of the points of the sample space S of the experiment E : { HHH, HTT, TTT, HHT, TTH, HTH, THH}. Find the missing
1.2.6 John and Mary plan to meet each other. Each of them is to arrive at the meeting place at some time between 5 p.m. and 6 p.m. John is to wait 20 minutes (or until 6 p.m., whichever comes first), and then leave if Mary does not show up. Mary will wait only 5 minutes (or until 6 p.m., whichever
1.2.5 The experiment consists of placing six balls in three boxes. List all outcomes in the sample space if (i) The balls are indistinguishable, but the boxes are distinguishable.(Hint: There are 28 different placements.) (ii) Neither the balls nor the boxes are distinguishable. (iii) Two balls are
1.2.4 In a game of craps, the player rolls a pair of dice. If he gets a total of 7 or 1 1, he wins at once; if the total is 2, 3, or 12, he loses at once. Otherwise, the sum, say 5, is his “point,” and he keeps rolling dice until either he rolls another 3: (in which case he wins) or he rolls a
1.2.3 An urn contains five chips, labeled 1, . . . , 5 . Three chips are drawn. List all outcomes included in the event “the second largest number drawn was 3.”
1.2.2 Alice, Bob, Carl, and Diana enter the elevator on the first floor of a four-story building. Each of them leaves the elevator on either the second, third, or fourth floor.(i) Find a simple way of describing the sample space (do not list all sample points).(ii) List all sample points such that
1.2.1 List all sample points in sample spaces for the following experiments: (i) We toss a coin. If heads come up, we toss a die. Otherwise, we toss the coin two more times. (ii) A coin is tossed until the total of two tails occurs, but no more than four times (i.e., a coin is tossed until the
Let X1, ...,Xn be a random sample from a POI(θ) distribution. (i) Using Jeffreys prior, find the posterior distribution for θ. (ii) Obtain the 95% credible interval forθ. (iii) Obtain the HDP 95% interval for θ if n = 3 and the actual observations are 2, 5, and 0. [Hint: The (1 − α)100%
Assume that X1, ...,Xn is a random sample from a BIN(1, θ) distribution and that the prior distribution of θ is U(0, 1). Find the Bayes estimator of θ if the loss function is L(θ, ˆθ)=(θ − ˆθ)2/[θ(1 − θ)]3/2.
Refer to Example 16.7. Find the Bayes estimator of θ after a was observed under(i) Squared error loss. (ii) Absolute error loss. (iii) Repeat (i) and (ii) assuming that both, a and d were observed.
Let X, ...,X16 be a random sample from a N(θ, 4) distribution and let the prior distribution of θ be N(0, 0.25). (i) Find the Bayes estimator of θ assuming squared error loss. (ii) Obtain a 95% credible interval for θ.
Let X, ...,Xn be a random sample from a GEO(θ) distribution and let the prior distribution of θ be BETA(2, 2). (i) Find the posterior distribution π(θ|x) and obtain the Bayes estimator of θ assuming squared error loss. (ii) Compare the obtained Bayes estimator with the MLE for the same data.
An insurance company estimates that about 40% of travelers (with the standard deviation of 20%) buy insurance for their airplane tickets. If in a random sample of 100 travelers who purchased an airplane tickets on a given day exactly 35 of them also purchased insurance, find the Bayes estimate of
Assume that X1, ...,X20 is a random sample from the N(θ, 1). (i) Obtain the Bayes factor for testing H0 : θ = 0 against H1 : θ = 0 if the prior distribution of θ under H1 is N(0.5, 0.25). Use xi = 7. (ii) Test the same hypothesis in a non-Bayesian framework and compare both results.
Five people were playing a board gave in which to start the game one had to roll a die and obtain either a one or a six dots. It took them respectively 5, 3, 1, 4, and 2 rounds, to start the game. If we assume that the die that they used was fair and that the prior probability θ of getting either
A diastolic blood pressure in a certain population of patients has a N(μ, 81) distribution. If the blood pressure of 10 randomly selected patients is 76, 71, 82, 63, 64, 75, 81, 78, 66, 62 and μ has a prior distribution N(70, 1), find the predictive distribution of the systolic blood pressure of
Let X1, ...,Xn be a random sample from the EXP(θ) distribution, and let the prior distribution of θ be EXP(β). (i) Find the Bayes estimator of θ and of μ = 1/θusing the squared error loss. (ii) Obtain the 90% credible interval for θ and for μ if n = 6, x = 4, and β = 2.
A shipment of 100 batteries contains θ defectives. (i) If the prior distribution π(θ)is BIN(100, 0.03), find the posterior distribution π(θ|x) knowing that one item was randomly selected from the shipment, and was found to not be defective. (ii) Find the predictive probability that the next
Let X1, ...,Xn be a random sample from a N(θ, 1) distribution. Using an improper constant prior for parameter θ (see Example 16.9), show that the predictive distribution of the next observation y is N(x, 1+1/n).
Measure the association of GPA and attitude toward statistics courses based on the data from Problem 15.3.5 using γ,τ ˆb, and d coefficients.
A study reported in Science magazine investigated the relationship between sex, handedness (right- or left-handed), and relative foot size (left foot bigger than right foot, left foot within one-half shoe size of right foot, or right foot bigger). A random sample of 150 adults gave the following
Fifty randomly selected college students that take courses with on-line quizzes were classified according to their answer on two questions: “Do you cheat taking on-line quizzes?” and “Do other students that you know well cheat taking on-line quizzes?”Explain what can be said about
A study of change in employment status was conducted among residents of a certain county. A random sample of 100 adult residents was surveyed on their employment status, and their responses (Yes/No) were recorded. The same people were contacted after 12 months and asked the same question. Their
Two out of ten randomly selected men, and four out of ten randomly selected women were found to be allergic to some drug. (i) Does these data indicate that there is a difference between men and women in their propensity to develop an allergic reaction to the drug in question? (ii) Does these data

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Probability And Statistical Inference 3rd Edition Robert Bartoszynski, Magdalena Niewiadomska-Bugaj - Solutions

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