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mathematics
statistics

Questions and Answers of Statistics

A run is a maximal sequence of successes in a sequence of Bernoulli trials. For example, in the sequence S, S, S,F, S, S,F,F, S, where S represents success and F represents failure, there are three
What is the variance of the number of heads that come up when a fair coin is flipped 10 times?
What is the expected number of times a 6 appears when a fair die is rolled 10 times?
Let A(X) = E(| X − E(X) |), the expected value of the absolute value of the deviation of X, where X is a random variable. Prove or disprove that A(X + Y) = A(X) + A(Y) for all random variables X
Suppose that X1 and X2 are independent Bernoulli trials each with probability 1/2, and let X3 = (X1 + X2) mod 2. a) Show that X1, X2, and X3 are pair wise independent, but X3 and X1 + X2 are not
Use Chebyshev's inequality to find an upper bound on the probability that the number of tails that come up when a fair coin is tossed n times deviates from the mean by more than 5√n.
Let X be a random variable on a sample space S such that X(s) ≥ 0 for all s ∈ S. Show that p(X(s) ≥ a) ≤ E(X)/afor every positive real number a. This inequality is called Markov's inequality.
Suppose that the number of tin cans recycled in a day at a recycling center is a random variable with an expected value of 50,000 and a variance of 10,000. a) Use Markov's inequality (Exercise 37) to
In this exercise we derive an estimate of the average-case complexity of the variant of the bubble sort algorithm that terminates once a pass has been made with no interchanges. Let X be the random
Show that V (X + Y) = V (X) + V (Y) + 2 Cov(X, Y).
When m balls are distributed into n bins uniformly at random, what is the probability that the first bin remains empty?
What is the expected number of bins that remain empty when m balls are distributed into n bins uniformly at random?
What is the expected sum of the numbers that appear on two dice, each biased so that a 3 comes up twice as often as each other number?
The final exam of a discrete mathematics course consists of 50 true/false questions, each worth two points, and 25 multiple-choice questions, each worth four points. The probability that Linda
Suppose that the probability that x is in a list of n distinct integers are 2/3 and that it is equally likely that x equals any element in the list. Find the average number of comparisons used by the
What is the probability that six consecutive integers will be chosen as the winning numbers in a lottery where each number chosen is an integer between 1 and 40 (inclusive)?
Suppose that a fair standard (cubic) die and a fair octahedral die are rolled together. a) What is the expected value of the sum of the numbers that come up? b) What is the variance of the sum of the
Suppose n people, n ≥ 3, play "odd person out" to decide who will buy the next round of refreshments. The n people each flip a fair coin simultaneously. If all the coins but one come up the same,
Suppose that m and n are positive integers. What is the probability that a randomly chosen positive integer less than mn is not divisible by either m or n?
There are three cards in a box. Sides of one card are b1ack, both sides of one card are red, and the third card has one b1ack side and one red side. We pick a card at random and observe on1y one
What is the probability that a randomly selected bit string of length 10 is a palindrome?
Consider the following game. A person flips a coin repeatedly until a head comes up. This person receives a payment of 2n dollars if the first head comes up at the nth flip. a) Let X be a random
Suppose that A and B are events with probabilities p(A) = 3/4 and p(B) = 1/3. a) What is the largest p(A ∩ B) can be? What is the smallest it can be? Give examples to show that both extremes for
a) Solve this puzzle in two different ways. First, answer the problem by considering the probability of the gender of the second child. Then, determine the probability differently, by considering the
Let X be a random variable on a sample space S. Show that V (aX + b) = a2V (X) whenever a and b are real numbers.
A player in the Powerball lottery picks five different integers between 1 and 59, inclusive, and a sixth integer between 1 and 39, which may duplicate one of the earlier five integers. The player
Suppose that at least one of the events Ej , j = 1, 2, . . . , m, is guaranteed to occur and no more than two can occur. Show that if p(Ej) = q for j = 1, 2, . . . , m and p(Ej ∩ Ek) = r for 1 ≤
What is the probability that each player has a hand containing an ace when the 52 cards of a standard deck are dealt to four players?
What is the probability that a 13-card bridge hand contains a) All 13 hearts? b) 13 cards of the same suit? c) Seven spades and six clubs? d) Seven cards of one suit and six cards of a second
a) What is the expected value of the number that comes up when a fair octahedral die is rolled? b) What is the variance of the number that comes up when a fair octahedral die is rolled?
Suppose that a pair of fair octahedral dice is rolled. a) What is the expected value of the sum of the numbers that come up? b) What is the variance of the sum of the numbers that come up?
Given a positive integer n, generate a random permutation of the set {1, 2, 3, . . . , n}.
Given positive integers m and n, generate m random permutations of the first n positive integers. Find the number of inversions in each permutation and determine the average number of these
Show that if R(n) is the number of moves used by the Frame-Stewart algorithm to solve the Reve's puzzle with n disks, where k is chosen to be the smallest integer with n ≤ k(k + 1)/2, then R(n)
a) Find a recurrence relation for the number of ways to lay out a walkway with slate tiles if the tiles are red, green, or gray, so that no two red tiles are adjacent and tiles of the same color are
Construct a variation of the algorithm described in Example 12 along with justifications of the steps used by the algorithm to find the smallest distance between two points if the distance between
How many rounds are in the elimination tournament described in Exercise 14 when there are 32 teams?
Suppose that f (n) = f (n/5) + 3n2 when n is a positive integer divisible by 5, and f (1) = 4. Find a) f (5). b) f (125). c) f (3125).
How many comparisons are needed for a binary search in a set of 64 elements?
Find the generating function for the finite sequence 2, 2, 2, 2, 2, 2.
Show that the coefficient pd (n) of xn in the formal power series expansion of (1 + x) (1 + x2) (1 + x3) ··· equals the number of partitions of n into distinct parts, that is, the number of ways
A survey of households in the United States reveals that 96% have at least one television set, 98% have telephone service, and 95% have telephone service and at least one television set. What
There are 2504 computer science students at a school. Of these, 1876 have taken a course in Java, 999 have taken a course in Linux, and 345 have taken a course in C. Further, 876 have taken courses
Find the number of positive integers not exceeding 100 that are either odd or the square of an integer.
How many elements are in the union of four sets if the sets have 50, 60, 70, and 80 elements, respectively, each pair of the sets has 5 elements in common, each triple of the sets has 1 common
Find a formula for the probability of the union of n events in a sample space.
Suppose that in a bushel of 100 apples there are 20 that have worms in them and 15 that have bruises. Only those apples with neither worms nor bruises can be sold. If there are 10 bruised apples that
Find the number of primes less than 200 using the principle of inclusion-exclusion.
How many solutions does the equation x1 + x2 + x3 = 13 have where x1, x2, and x3 are nonnegative integers less than 6?
Find the solution to the recurrence relation f (n) = f (n/2) + n2 for n = 2k where k is a positive integer and f (1) = 1.
Give a big-O estimate for the number of comparisons used by the algorithm described in Exercise 22.
Show that the algorithm from Exercise 24 has worst-case time complexity O(log n) in terms of the number of comparisons.
Let [αn] and [bn] be sequences of real numbers. Show that ∆(αnbn) = αn+1(∆bn) + bn(∆an).
How many bit strings of length six do not contain four consecutive 1s?
What is the probability that a bit string of length six chosen at random contains at least four 1s?
List the ordered pairs in the relation R from A = {0, 1, 2, 3, 4} to B = {0, 1, 2, 3}, where (a, b) ∈ R if and only if a) a = b. b) a + b = 4. c) a > b. d) a | b. e) gcd(a, b) = 1. f) lcm(a, b) = 2.
Which relations in Exercise 3 are irreflexive? A relation R on the set A is irreflexive if for every a ∈ A, (a, a) ∉ R. That is, R is irreflexive if no element in A is related to itself.
Which relations in Exercise 4 are asymmetric? A relation R is called asymmetric if (a, b) ∈ R implies that (b, a) ∉ R. Explore the notion of an asymmetric relation. Exercise 22 focuses on the
Let A be the set of students at your school and B the set of books in the school library. Let R1 and R2 be the relations consisting of all ordered pairs (a, b), where student a is required to read
Let R be the relation on the set of people consisting of pairs (a, b), where a is a parent of b. Let S be the relation on the set of people consisting of pairs (a, b), where a and b are siblings
Let R be the relation on the set of people with doctorates such that (a, b) ∈ R if and only if a was the thesis advisor of b. When is an ordered pair (a, b) in R2? When is an ordered pair (a, b) in
How many of the 16 different relations on {0, 1} contain the pair (0, 1)?
a) How many relations are there on the set {a, b, c, d}? b) How many relations are there on the set {a, b, c, d} that contain the pair (a, a)?
Find the error in the "proof" of the following "theorem." "Theorem": Let R be a relation on a set A that is symmetric and transitive. Then R is reflexive. "Proof ": Let a ∈ A. Take an element b ∈
Show that the relation R on a set A is symmetric if and only if R = R−1, where R−1 is the inverse relation.
Show that the relation R on a set A is reflexive if and only if the inverse relation R−1 is reflexive.
Let R be a relation that is reflexive and transitive. Prove that Rn = R for all positive integers n.
Let R be a reflexive relation on a set A. Show that Rn is reflexive for all positive integers n.
Suppose that the relation R is irreflexive. Is R2 necessarily irreflexive? Give a reason for your answer.
What do you obtain when you apply the selection operator sC, where C is the condition Destination = Detroit, to the database in Table 8?
What do you obtain when you apply the selection operator sC, where C is the condition (Airline = Nadir) ˆ¨ (Destination = Denver), to the database in Table 8?
Which projection mapping is used to delete the first, second, and fourth components of a 6-tuple?
Display the table produced by applying the projection P1,4 to Table 8.
Construct the table obtained by applying the join operator J2 to the relations in Tables 9 and 10.Table 9Table 10
Show that if C1 and C2 are conditions that elements of the n-ary relation R may satisfy, then sC1 (sC2(R)) = sC2 (sC1(R)).
Show that if C is a condition that elements of the n-ary relations R and S may satisfy, then sC(R ∩ S) = sC(R) ∩ sC(S).
Show that if R and S are both n-ary relations, then Pi1,i2,...,im(R ∪ S) = Pi1,i2,...,im(R) ∪ Pi1,i2,...,im(S).
Give an example to show that if R and S are both n-ary relations, then Pi1,i2,...,im(R − S) may be different from Pi1,i2,...,im(R) − Pi1,i2,...,im(S).
a) What are the operations that correspond to the query expressed using this SQL statement? SELECT Supplier, Project FROM Part_needs, Parts_inventory WHERE Quantity ≤ 10 b) What is the output of
List the 5-tuples in the relation in Table 8.
Determine whether there is a primary key for the relation in Example 3.
Assuming that no new n-tuples are added, find a composite key with two fields containing the Airline field for the database in Table 8.
The 3-tuples in a 3-ary relation represent the following attributes of a student database: student ID number, name, phone number. a) Is student ID number likely to be a primary key? b) Is name likely
The 5-tuples in a 5-ary relation represent these attributes of all people in the United States: name, Social Security number, street address, city, state. a) Determine a primary key for this
Represent each of these relations on {1, 2, 3} with a matrix (with the elements of this set listed in increasing order). a) {(1, 1), (1, 2), (1, 3)} b) {(1, 2), (2, 1), (2, 2), (3, 3)} c) {(1, 1),
How can the matrix for , the complement of the relation R, be found from the matrix representing R, when R is a relation on a finite set A?
Let R be the relation represented by the matrixFind the matrix representing a) Rˆ’1 b) R. c) R2.
Let R be the relation represented by the matrixFind the matrices that represent a) R2. b) R3. c) R4.
Let R be a relation on a set A with n elements. If there are k nonzero entries in MR, the matrix representing R, how many nonzero entries are there in M, the matrix representing R, the complement of
Draw the directed graphs representing each of the relations from Exercise 2.
Draw the directed graph representing each of the relations from Exercise 4.
How can the directed graph of a relation R on a finite set A be used to determine whether a relation is asymmetric?
List the ordered pairs in the relations on {1, 2, 3} corresponding to these matrices (where the rows and columns correspond to the integers listed in increasing order).a)b) c)
Determine whether the relations represented by the directed graphs shown in Exercises 23-25 are reflexive, irreflexive, symmetric, antisymmetric, and/or transitive.
Let R be a relation on a set A. Explain how to use the directed graph representing R to obtain the directed graph representing the inverse relation R−1.
Show that if MR is the matrix representing the relation R, then M[n]R is the matrix representing the relation Rn.
How can the matrix representing a relation R on a set A be used to determine whether the relation is irreflexive?
How many nonzero entries does the matrix representing the relation R onA = {1, 2, 3, . . . , 100} consisting of the first 100 positive integers have if R is a) {(a, b) | a > b}? b) {(a, b) | a ≠
Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0, 1), (1, 1), (1, 2), (2, 0), (2, 2), and (3, 0). Find the a) Reflexive closure of R. b) Symmetric closure of R.
Find the directed graph of the smallest relation that is both reflexive and symmetric that contains each of the relations with directed graphs shown in Exercises 5-7.In exercise1.2.

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