An Elementary Introduction To Statistical Learning Theory 1st Edition Sanjeev Kulkarni - Solutions

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A fair coin is tossed until a head appears. Given that the first head appeared on an even-numbered toss, what is the probability that it appeared on the 2nth toss?
A deck of eight cards contains four jacks and four queens. A pair of cards is drawn at random. Find the probability that both cards are jacks if: (i) At least one of the cards is a jack. (ii) At least one of the cards is a red jack. (iii) One of the cards is a jack of hearts.
Three distinct integers are chosen at random from the set {1, 2, ..., 15}. Find the probability that: (i) Their sum is odd. (ii) Their product is odd. (iii) The sum is odd if it is known that product is odd. (iv) The product is odd if it is known that the sum is odd. (v) Answer questions (i)–(iv)
Events A,B, and C are such that at least one of them occurs. Moreover, P(A|B) = P(B|C) = P(C|A)=1/2,P(B ∩ C)=2P(A ∩ B)=4P(C ∩ A)/3 =2P(A ∩ B ∩ C). Find the probability that: (i) Exactly one of events A,B, and C occurs. (ii) Only B occurs.
Find P[A ∩ B ∩ C|(A ∩ B) ∪ (A ∩ C)] if P(A)=0.8,P(B)=0.4, P(C)=0.4,P(A ∪ B)=1,P(A ∪ C)=0.9, and P(B ∪ C)=0.6.
Find P(A|B), if P(Ac)=2P(A) and P(B|Ac)=2P(B|A)=0.2.
P(A) = P(B)=1/2,P(A|B)=1/3. Find P(A ∩ Bc).
Find P(A ∩ B) if P(A)=3/4,P(B)=1/2,P(A|B) − P(B|A)=2/9.
Assume that P(A ∩ B) > 0 and determine P(A)/[P(A) + P(B)] as a function of a, where a = P(B|A)/P(A|B).
Assume that A and B are disjoint. Find P(Ac|B) and P(A ∪ B|Ac).
Let events A,B, and C be such that P(A) > 0,P(B) > 0, and P(C) > 0.Label the following statements as true or false: (i) The conditional probability P(A|B) can never exceed the unconditional probability P(A). (ii)If P(A|B) = P(A|C), then P(B) = P(C). (iii) If A and B are disjoint, then P(A|A ∪ B)
Approximate the probability that every state will be represented in the committee in Problem
Use Stirling’s formula to approximate the number of ways: (i) A set of size 2n can be partitioned into two equal parts. (ii) A set of size 3n can be partitioned into three equal parts.
to show that if i ≥ 1, j ≥ 1, and k ≥ 1, then n + 1 i, j, k=n i − 1, j, k+n i, j − 1, k+n i, j, k − 1.
Use the argument analogous to that in Theorem
Find the coefficient of the term x4y5z3 in the expansion of (x − 2y + 3z)12.
Show that if a ≤ b ≤ c ≤ n, thenn c c b b a=n a n − a b − a n − b c − b.(i) Use the definition of binomial coefficients as ratios of the factorials. (ii) Use directly the interpretation of the binomial coefficients as the number of subsets of a given size.(iii) How many ways can
Find the number of polygonal lines (as in Problem 3.3.12) that join the points (2,3)and (16, 5) and: (i) Never touch the x-axis. (ii) Never touch the line y = 7.
Find the number of polygonal lines with vertices (x, C(x)), where C(x) is as in Example 3.12 and with possible edges leading from (x, C(x)) to (x + 1, C(x) + 1) or(x + 1, C(x) − 1), connecting the points: (i) (0, 0) and (10, 0). (ii) (0, 0) and (10, 5).iii) (3, −2) and (8, 1).
Compute probabilities P(x) of winning x numbers in lotteries, where the player chooses: (i) 5 out of 44 numbers. (ii) 6 out of 55 numbers.
(i) If n balls are put at random into n boxes, find the probability of exactly one box remaining empty? (ii) If n balls are randomly placed into k boxes (n ≥ k), labeled 1, ...,k, find the probability that no box is empty.
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-a-kind.(ii) Four-of-a-kind. (iii) A full house.
A poker player has 3♥, 7♦, 8♦, 9♦, Q♣. She discards 3♥ and Q♣ 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.
A poker player has 3♥, 7♦, 8♠, 9♦, Q♣. He discards 3♥ and Q♣ and obtains 2 cards.8(i) What is the probability that he will have a straight? (ii) Answer the same question if Q♣ is replaced by J♣ (i.e., he discards 3♥ and J♣).
Find the probability that a poker hand will contain two pairs (one red and the other black) and one unmatched card.
(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 flush), (iv)
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
How many ways can one order the deck of 52 cards so that all four kings are next to each other?
A shipment of 30 items is received. For the quality control, three items are randomly selected, and if more than one of them is defective then the whole shipment is rejected.Find the probability that the shipment will be accepted if it has: (i) Three defective items (ii) Ten defective items.
(i) A committee of size 50 is to be formed out of the US Senate at random. Find the probability that each state will be represented. (ii) If a committee of size k is to be formed out of the US Senate find how large must k be in order for the event “at least one senator from Ohio is included” to
Let er be the probability that exactly two people in a group of r have the same birthday, and let pr be the probability that everybody in the group has a different birthday.Show that
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?
A regular die is tossed n times. Find the probability that: (i) Each side turns up exactly once if n = 6. (ii) Each side turns up at least once if n = 8.
Find the percentage of six-digit numbers that have all digits distinct.
A restaurant menu has 5 appetizers, 3 soups, 15 entrees, and 3 desserts. (i) Assuming you are going to order one item from each group, how many possible dinners can you order? (ii) Assume you are at the restaurant with a friend. How many different orders for two full dinners can you place if your
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 hat
A total of 12 girls and 17 boys go to a dance. (i) How many possible dancing pairs(boy–girl) may be formed? (ii) The dance floor can accommodate at most 11 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
Five men and five women are to be seated in a row of 10 chairs. Find the number of possible arrangements if: (i) Men are required to sit in alternating seats. (ii) No two men are to be seated next to each other.
Seated at random at a round table with n seats are n people, among them John and Mary. (i) Answer questions (i)–(iv) 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 the
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.
Determine the number of 0s at the end of 16! and 27!.
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.
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 of the 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 compute
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.
Suppose that Tom is confronted with the choice between two options: O1, which is simply to receive $1,000,000, or O2, 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, Tom
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 a situation, should they both agree that Peter buys the ticket from Tom for $150?
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 ticket if all
Use formula (2.6) 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 i.” Determine p = P(A2 ∪ A3 ∪ A5 ∪ A7). Then the answer to the problem is 100(1 − p)+3 (why?).]
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 10?
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 ).
A coin is tossed seven 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 0111001 = 57. Find the probability that a number generated in this
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?
Based on a survey of undergraduate sophomores at a certain university, the researchers found out that 67% of students live in a dorm (D), 52% are interested in studying abroad (A), and 48% are planning to attend graduate school (G). Furthermore, 32% live in a dorm and are planning studying abroad,
Before the era of cell phones (a time that most of today’s students do not remember), people often used public phones that operated after a coin was inserted. These phones were sometimes faulty and either took a coin without giving a connection or gave the connection but also returned the coin.
If events A1, ...A101 are such that P(A1) = ··· = P(A101)=0.01, P(A1 ∩ A2) =P(A1 ∩ A3) = ··· = P(A100 ∩ A101) = r, while every triple intersection is empty, find the smallest possible value of the probability of intersection r.
Let A, B, C, D, and E be five events such that P(A) = P(B) = ··· = P(E) = k and P(A ∩ B) = P(A ∩ C) = ··· = P(D ∩ 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. (ii)
Three events A, B, and C are such that C is contained in the intersection Ac ∩ B,P(A)=0.6, P(A ∩ B)=0.3, and P(C) = P(Ac ∩ Bc)=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 occur.
To make the formula P[(Ac ∩ Bc)c]=2 − P(Ac) − P(Bc) − x valid, x must equal:(1) P(A ∪ B). (2) P(A ∩ B). (3) P(Ac ∪ Bc). (4) None of the above.
Find P(A ∪ Bc) for events A and B such that P(A) = P(B)=1/2, and P(Ac ∩ Bc)=1/3.
Events A, B, and C are such that A ∩ B = A ∩ C = ∅ while B ∩ C is not empty. Determine P(A ∪ B ∪ C) as a function of x if P(A) = P(B) = P(C) =3P(B ∩ C) = x.
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 Ac. Find P(B ∪ C).
Show that first and third, as well as second and third, schemes of sampling chords (see Bertrand’s paradox) are not equivalent.
A bathroom floor is covered by square tiles with side lengtha. You drop a coin with diameterb, where b
Label all statements below as true or false. (i) If A is more likely to occur than Ac, then P(A) > 0.5. (ii) If A occurs whenever B does, then P(B) ≤ P(A).(iii) If P(A) ≤ P(B), then whenever B occurs, A does also. (iv) If P(A)=0.75, then A must occur three times out of every four. (v) The sum
Let S = (−∞, +∞). 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,∞), etc.) forms a field.
Let S = {0, 1, 2, ... } be the set of all integers. For A ⊂ S, let fn(A) be the number of elements in the intersection A ∩ {0, 1, ...,n}. Let A be the class of all sets A for which the limit exists. Show that A is not a field. [Hint: Let A1 = {1, 3, 5, ... } and A2 ={ all odd integers between
For 0
Let A1, ...,An be distinct subsets of S. (i) Find the maximum number of sets (including S and ∅) of the smallest field containing A1, ...,An. (ii) Find the maximum number of sets in this field if An−1 ⊂ An. (iii) Answer (ii) if A1 ⊂ A2 ⊂···⊂ An.(iv) Answer (ii) if A1 = ··· = An =
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, 2035. Label the following statements as true or false:(a) B = lim inf An. (b) B ⊂ lim inf An.
Let B1, B2 , ... be a countable partition of S; that is, Bi ∩ Bj = ∅ for all i= j, and iBi = S. Let An = Bn ∪ Bn+1 ∪···. Find lim An.
Let Anc be defined inductively by A0c = A, A(n+1)c = (Anc)c. Find Amc ∩ Anc and Amc ∪ Anc for m, n > 0.
Five burglars, A, B, C, D, and E, divide the loot, consisting of five identical gold bars and four identical diamonds. Let Ajk be the event that A got at least j gold bars and at most k diamonds. Let Bjk, Cjk denote analogous events for burglars B,C (e.g., B21 is the event that B got 2, 3, 4, or 5
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 Sk, Ek, and Wk be defined similarly. For each of the events below, determine the number of aces that N has. (i) N1 ∩ S1 ∩ E1 ∩W1. (ii) E2 ∩ (W2 ∪ S2).
Let A, B, and C be three events. Match, where possible, events D1 through D10 with events E1 through E11. 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, ...,D10,
In a group of 1,000 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 the
Find X if: (i) A ÷ X = ∅. (ii) A ÷ X = A. (iii) A ÷ X = S. (iv) A ÷ X = B.
In the statements below A, B, C, and D are events. Find those statements or formulas that are true. (i) If A ∩ B = A ∩ C, then B = C. (ii) A ∪ (A ∩ B) ∪ (A ∩ Bc) = A.(iii) A ∪ (A ∩ B) ∪ (A ∩ Bc) = B. (iv) If A\B = C, then A = B ∪ C. (v) (A ∪ B) ∩(C ∪ D)=(A ∩ C) ∪ (A
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 are
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 observes
Let E be an experiment consisting of tossing a die twice. Let S be the sample space with sample points (i, j), i, j = 1, 2, ..., 6, with i and j being the numbers of dots that appear in the first and second toss, respectively.
Let E 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, HTH, THH}. Use a tree diagram to find the
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 comes
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
In a game of craps, the player rolls a pair of dice. If he gets a total of 7 or 11, he wins at once; if the total is 2, 3, or 12, he loses at once. Otherwise, the sum, say x, is his“point,” and he keeps rolling dice until either he rolls another x (in which case he wins) or he rolls a 7 in
An urn contains five chips, labeled 1, ..., 5. Three chips are selected. List all outcomes included in the event “the second largest number drawn was 3.”
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) Describe the sample space without listing all sample points. (ii) List all sample points such that Carl and Diana leave the
List all sample points in sample spaces for the following experiments: (i) We toss a balanced coin.1 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
Consider the following method for using data to come up with a good classification rule: use the data to estimate the background probability distributions and then use these distributions to find a Bayes rule. How does this method compare with other learning methods discussed in this book?
Philosophers sometimes suggest that there is a sense in which you are justified in believing something only if you believe it as the result of a reliable process of belief formation. What do you think they mean? Is there anything about statistical learning theory that might help explain what it is
How can boosting turn a weak classifier into a better classifier?
If ht(x) is the rule produced by boosting at round t, write the expression for the final classifier H (x) after T rounds in terms of the ht(x) and the weights αt .
True or false: Boosting is best seen as a special case of a kernel method.
What is boosting? What is a “weak” learning rule? When might boosting be a useful method of improving the results of weak learning rules?
Suppose that in the transformed space the feature vectors and corresponding labels are given as follows:(−1, 0; +), (0, 1; +), (1, 0; +), (−1, 2; −), (−1, 3; −), (0, 3; −), (1, 5; −)(a) Plot the feature vectors. Label the support vectors and solve for the margin.What is the SVM
Why are SVMs called support vector machines. What is a support vector and what role does it play in an SVM?
SVMs solve a classification problem by, in effect, transforming the feature space into another typically higher dimensional, sometimes infinitely dimensional space and then finding a (possibly soft) wide margin linear separation in that other space. What is the VC-dimension of linear
How does the construction of the previous problem affect the margin of the classifier?
Show that by increasing the dimension of the feature vector by 1, a hyperplane passing through the origin in the augmented space can represent a general hyperplane decision rule in the original space.
If we insist that the hyperplane pass through the origin, what will the form of the rule be?
What is the form of the decision rule for a linear SVM?

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An Elementary Introduction To Statistical Learning Theory 1st Edition Sanjeev Kulkarni - Solutions

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