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linear algebra
Questions and Answers of
Linear Algebra
Suppose that A, B are independent with Pr(A U B) = 0.6 and Pr(A) = 0.3. Find Pr(B).
Let X be a random variable with the following probability distribution.Determine(a) Pr(X = 3).(b) Pr(X (c) Pr(X > 0).(d) Pr(1 (e) Pr(X = 2|X (f) Pr(Z
A carnival game invites a player to select one card from a standard deck of 52 cards. If the card is a seven or a jack the player is given five dollars. For a king or an ace the player is given eight
The route that Jackie follows to school each day includes eight stoplights. When she reaches each stoplight, the probability that the stoplight is red is 0.25 and it is assumed that the stoplights
Suppose that a random variable Z has mean E(X) = 17 and variance Var(X) = 9, but its probability distribution is unknown. Use Chebyshev's Inequality to estimate a lower bound for (a) Pr(11 < Z <
Suppose that a random variable Z has mean E(X) = 15 and variance Var(X) = 4, but its probability distribution is unknown. Use Chebyshev's Inequality to find the value of the constant c where Pr(|X -
Fred rolls a fair die 20 times. If Z is the random variable that counts the number of 6's that come up during the 20 rolls, determine E(X) and Var(X).
A carton contains 20 computer chips, four of which are defective. Isaac tests these chips - one at a time and without replacement - until he either finds a defective chip or has tested three chips.
Suppose that X is a random variable defined on a sample space and that a, b are constants. Show that (a) E(aX + b) = aE(x) + b. (b) Var(aX + b) = a2Var(X).
Let X be a binomial random variable with Pr(X = x) = (n/r)pXqn-x, x - 0, 1 2, . . ., n, where n (> 2) is the number of Bernoulli trials, p is the probability of success for each trial, ad q = 1 -
In alpha testing a new software package, a software engineer finds that the number of defects per 100 lines of code is a random variable X with probability distribution:Find (a) Pr(X > 1). (b) Pr(X =
In Mario Puzo's novel The Godfather, at the wedding reception for his daughter Constanzia, Don Vito Corleone discusses with his godson Johnny Fontane how he will deal with the movie mogul Jack Woltz.
The probability distribution for a random variable X is given by Pr(X = x) = (3x + l)/22, x = 0, 1, 2, 3. Determine (a) Pr(X = 3). (b) Pr{X < 1). (c) Pr(l < X < 3). (d) Pr(X > -2). (e) Pr(X = 1\X <
An assembly comprises three electrical components that operate independently. The probabilities that these components function according to specifications are 0.95, 0.9, and 0.88. If the random
An urn contains five chips numbered 1, 2, 3, 4, and 5. When two chips are drawn (without replacement) from the urn, the random variable X records the higher value. Find E (X) and σx.
A shipment of 120 graphics cards contains 10 that are defective. Serena selects five of these cards, without replacement, and inspects them to see which, if any, are defective. If the random variable
Connie tosses a fair coin three times. If X = X1 - X2, where X1 counts the number of heads that result and X2 counts the number of tails that result, determine(a) The probability distributions for
Let X be the random variable where Pr(X = x) = 1/6 for x = 1, 2, 3, .. ., 6. (Here X is a uniform discrete random variable.) Determine (a) Pr(X > 3). (b) Pr(2 < X < 5); (c) Pr(X = 4|X > 3). (d)
A computer dealer finds that the number of laptop computers her dealership sells each day is a random variable Z where the probability distribution for Z is given bywhere c is a constant.
A random variable Z has probability distribution given bywhere c is a constant. Determine (a) The value of c. (b) Pr(X (c) E(X). (d) Var(X).
Wayne tosses an unfair coin - one that is biased so that a head is three times as likely to occur as a tail. How many heads should Wayne expect to see if he tosses the coin 100 times?
Suppose that X is a binomial random variable where Pr(X = x) = (n/x)px(l - p)n-x, x = 0, 1, 2, . . ., n. If E(X) = 70 and Var(X) = 45.5, determine n, p.
Let A, B, C ⊂ U. Prove that (A - B) ⊂ C if and only if (A - C) ⊂ B.
Let U be a given universe with A, B Š‚ U, |A ˆ© B| = 3, |A U B| = 8, and |U| = 12.a) How many subsets C Š‚ U satisfy A ˆ© B Š‚ C Š‚ A U B? How many of these subsets C contain an even
Let U = R and let the index set I = Q+. For each q Q+, let Aq = [0, 2q] and Bq = (0, 3q]. Determinea) A7/3b) A3 B4c)d)
For a universe U and sets A, B Š‚ U, prove thata) A ˆ† B = B ˆ† Ab)c)d) A ˆ† θ = A, so θ is the identity for A, as well as for U
Consider the membership table (Table 3.7). If we are given the condition that A B, then we need consider only those rows of the table for which this is true - rows 1, 2, and 4, as
State the dual of each theorem in Exercise 13. (Here you will want to use the result of Example 3.19 in conjunction with Theorem 3.5.)In Exercise 13Consider the membership table (Table 3.7). If we
a) Determine the number of linear arrangements of m l's and r 0's with no adjacent l's. (State any needed condition(s) for m, r.)b) If U = {1, 2, 3, ..., n}, how many sets A ⊂ U are such that | A |
In how many ways can 15 laboratory assistants be assigned to work on one, two, or three different experiments so that each experiment has at least one person spending some time on it?
Give a combinatorial argument to show that for integers n, r with n > r > 2,
Professor Diane gave her chemistry class a test consisting of three questions. There are 21 students in her class, and every student answered at least one question. Five students did not answer the
Let U be a given universe with A, B ⊂ U, A ∩ B = θ, |A| = 12, and |B| = 10. If seven elements are selected from A U B, what is the probability the selection contains four elements from A and
For a finite set A of integers, let a (A) denote the sum of the elements of A. Then if U is a finite universe taken from Z+, ∑A∈p(U) σ(A) denotes the sum of all elements of all subsets of U.
a) In chess, the king can move one position in any direction. Assuming that the king is moved only in a forward manner (one position up, to the right, or diagonally northeast), along how many
Let A, B ⊂ R, where A = {x|x2 - 7x = -12} and B = {x|x2 - x = 6}. Determine A U B and A ∩ B.
Let A, B ⊂ R, where A = {x|x2 - 7x < -12} and B = {x|x2 - x < 6}. Determine A U B and A ∩ B.
Four torpedoes, whose probabilities of destroying an enemy ship are 0.75, 0.80, 0.85, and 0.90, are fired at such a vessel. Assuming the torpedoes operate independently, what is the probability the
Travis tosses a fair coin twice. Then he tosses a biased coin, one where the probability of a head is 3/4, four times. What is the probability Travis's six tosses result in five heads and one tail?
Let S be the sample space for an experiment ε, with events A, B Š‚ Prove that
Let A, B, C be independent events taken from a sample space S. Prove that the events A and B U C are independent.
Let A, B, ⊂ U. Prove or disprove (with a counterexample) each of the following:a) A - C = B - C ⇒ A = Bb) [(A ∩ C = B ∩ C) ∧ (A - C = B - C)] ⇒ A = Bc) [(A U C = B U C) ∧ (A - C = B -
What is the minimum number of times we must toss a fair coin so that the probability that we get at least two heads is at least 0.95?
A large jet aircraft has two wheels per landing gear for added safety. The tires are rated so that even with a "hard landing" the probability of any single tire blowing out is only 0.10. (a) What is
Let be the sample space for an experiment ε and let A, B be events - that is, A, B ⊂ S. Prove that Pr(A ∩ B) > Pr(A) + Pr(B) - 1. (This result is known as BonferronVs Inequality.)
The exit door at the end of a hallway is open half of the time. On a table by the entrance to this hallway is a box containing 10 keys, but only one of these keys opens the exit door at the end of
Dustin tosses a fair coin eight times. Given that his first and last outcomes are the same, what is the probability he tossed five heads and three tails?
Suppose that the number of boxes of cereal packaged each day at a certain packaging plant is a random variable - call it X-with E(X) = 20,000 boxes and Var(X) = 40,000 boxes2. Use Chebyshev's
Find the probability of getting one head (exactly) two times when three fair coins are tossed four times.
Devon has a bag containing 22 poker chips - eight red, eight white, and six blue. Aileen reaches in and withdraws three of the chips, without replacement. Find the probability that Aileen has
Let X be a random variable with probability distributionwhere c is a constant. Determine (a) the value of c. (b) Pr(X > 1). (c) Pr(X = 3|X > 2). (d) E(X). (e) Var(X).
a) For positive integers m, n, r, with rb) For n a positive integer, show that
A dozen urns each contain four red marbles and seven green ones. (All 132 marbles are of the same size.) If a dozen students each select a different urn and then draw (with replacement) five marbles,
Maureen draws five cards from a standard deck: the 6 of diamonds, 7 of diamonds, 8 of diamonds, jack of hearts, and king of spades. She discards the jack and king and then draws two cards from the
A grab bag contains one chip with the number 1, two chips each with the number 2, three chips each with the number 3, ..., and n chips each with the number n, where n e Z+. All chips are of the same
A fair die is rolled three times and the random variable X records the number of different outcomes that result. For example, if two 5's and one 4 are rolled, then X records two different outcomes.
When a coin is tossed three times, for the outcome HHT we say that two runs have occurred - namely, HH and T. Likewise, for the outcome THT we find three runs: T, H, and T. (The notion of a run was
Determine whether each of the following statements is true or false. For each false statement, give a counterexample. a) If A and B are infinite sets, then A ∩ B is infinite. b) If B is infinite
Let A = {1, 2, 3, ... , 15}.a) How many subsets of A contain all of the odd integers in A?b) How many subsets of A contain exactly three odd integers?c) How many eight-element subsets of A contain
Let A, B, C ⊂ U. Prove that (A ∩ B) U C = A ∩ (B U C) if and only if C ⊂ A.
Prove each of the following for all n > 1 by the Principle of Mathematical Induction.(a) 12 + 32 + 52 + + (2n - l)2 = n(2n - 1)(2n + 1)/3(b)
Determine ∑100i=51 t, where tt denotes the ith triangular number, for 51 < i < 100.
a) Derive a formula for ∑ni=1 t2l, where t2i denotes the 2ith triangular number for 1 < i < n. b) Determine ∑100i=1 t21. c) Write a computer program to check the result in part (b).
(a) Prove that (cos θ + i sin θ)2 = cos 2θ + i sin 2θ, where i ∈ C and i2 = -1. (b) Using induction, prove that for all n ∈ Z+, (cos θ + i sin θ)n = cos nθ + i sin nθ. (This result is
(a) Consider an 8 X 8 chessboard. It contains sixty-four 1 × 1 squares and one 8 × 8 square. How many 2 × 2 squares does it contain? How many 3 × 3 squares? How many squares in total? (b) Now
Prove that for all n ∈ Z+, n > 3 ⇒ 2n < n.
Prove that for all n ∈ Z+, n > 4 =⇒ n2 < 2n.
(a) For n = 3 let X3 = {1, 2, 3}. Now consider the sumwhere pA denotes the product of all elements in a nonempty subset A of X3. Note that the sum is taken over all the nonempty subsets of X3.
For n Z+, let Hn denote the nth harmonic number (as defined in Example 4.9).(a) For all n N prove that 1 + (n/2) (b) Prove that for all n Z+,
Consider the following four equations:1) 1 = 12) 2 + 3+ 4= 1 + 83) 5 + 6 + 7 + 8 + 9 = 8 + 274) 10 + 11 + 12 + 13 + 14 + 15 + 16 = 27 + 64Conjecture the general formula suggested by these four
For n Z+, let S(n) be the open statementShow that the truth of S(k) implies the truth of S (k + 1) for all k Z+. Is S(n) true for all n Z+?
Establish each of the following for all n > 1 by the Principle of Mathematical Induction.a.b. c.
Let S1 and S2 be two sets where |S1| = m, |S2| - r, for m, r ∈ Z+, and the elements in each of S1, S2 are in ascending order. It can be shown that the elements in S1 and S2 can be merged into
During the execution of a certain program segment (written in pseudocode), the user assigns to the integer variables x and n any (possibly different) positive integers. The segment shown in Fig. 4.8
In the program segment shown in Fig. 4.9, x, y, and answer are real variables, and n is an integer variable. Prior to execution of this while loop, the user supplies real values for x and y and a
(a) Let n ∈ Z+, where n ≠ 1, 3. Prove that n can be expressed as a sum of 2's and/or 5's. (b) For all n ∈ Z+ show that if n > 24, then n can be written as a sum of 5's and/or 7's.
A sequence of numbers a1, a2, a2, . . . is defined by a1 = 1 a2 = 2 an = an-1 + an-2, n > 3. (a) Determine the values of a3, α4, α5, α6, and a7. (b) Prove that for all n > 1, an < (7/4)n.
For a fixed n ∈ Z+, let X be the random variable where Pr(X = x) = 1/n, x = 1, 2, 3, . . . , n. (Here X is called a uniform discrete random variable.) Determine E(X) and Var(X).
a) Show that α1 = α20 and that α2 = 2α30b) Determine a3 and a4 in terms of a0.c) Conjecture a formula for αn in terms of
Verify Theorem 4.2. The Principle of Mathematical Induction-Alternative Form. Let S(n) denote an open mathematical statement (or set of such open statements) that involves one or more occurrences of
(a) Of the 25-1 = 24 = 16 compositions of 5, determine how many start with (i) 1. (ii) 2. (iii) 3. (iv) 4. (v) 5. (b) Provide a combinatorial proof for the result in part (a) of Exercise 2.
(a) Note how ∑ni=1 i3 + (n + l)3 = ∑ni=0 (i + 1)3 = ∑ni=0 (i3 + 3i + 1). Use this result to obtain a formula for ∑ni=1 i2. (Compare with the formula given in Example 4.4.) (b) Use the idea
A wheel of fortune has the integers from 1 to 25 placed on it in a random manner. Show that regardless of how the numbers are positioned on the wheel, there are three adjacent numbers whose sum is at
a) For the four-digit integers (from 1000 to 9999) how many are palindromes and what is their sum? b) Write a computer program to check the answer for the sum in part (a).
A lumberjack has 4n + 110 logs in a pile consisting of n layers. Each layer has two more logs than the layer directly above it. If the top layer has six logs, how many layers are there?
Determine the positive integer n for which
Evaluate each of the following: (a) ∑33i=1 i (b) ∑33i=1 i2.
The integer sequence a1, a2, a3, . . ., defined explicitly by the formula an = 5n for n e Z+, can also be defined recursively by 1) a1 = 5; and 2) an+1, an + 5, for n > 1. For the integer sequence
For all x R,-|x| Prove that if n Z+, n > 2, and x1, x2, . . . , R, then |x1 + x2 + + xn|
Define the integer sequence a0, < a1 > a2, a3, . . ., recursively by 1) a0 = 1 a1 = 1, a2 = 1; and 2) For n > 3, an = an-1 + an-3. Prove that an+2 > (√2)n for all n > 0.
For n > 0 let Fn denote the nth Fibonacci number. Prove that
Prove that for any positive integer n,
As in Example 4.20 let L0, L1, L2, . .. denote the Lucas numbers, where (1) L0 = 2, L1 = 1; and (2) Ln+2 = Ln+1 + Ln, for n > 0. When w > 1, prove thatL21 + L22 + L23 +∙ ∙ ∙ ∙ + =
If n ∈ N, prove that 5Fn+2 = Ln+4 - L.
Give a recursive definition for the set of all (a) Positive even integers (b) Nonnegative even integers
One of the most common uses for the recursive definition of sets is to define the well-formed formulae in various mathematical systems. For example, in the study of logic we can define the
Consider the permutations of 1, 2, 3, 4. The permutation 1432, for instance, is said to have one ascent - namely, 14 (since 1 < 4). This same permutation also has two descents - namely, 43 (since 4 >
(b) Fix n in Z+. Since the result in part (a) is true for all k = 1, 2, 3, ..., n, summing the n equations
(a) Give a recursive definition for the disjunction of statements p1 p2, . . . , pn, Pn+1 n > 1. (b) Show that if n, r ∈ Z+, with n > 3 and 1 < r < n, then (p1 ∨ p2 ∨ ∙ ∙ ∙ ∙ ∨ pr)
Use the result of Example 4.16 to prove that if p, q1, q2,. . . . . .qn are statements and n > 2, then P ∨ (q1 ∧ q2 ∧ ∙ ∙ ∙ ∧ qn) ⇔ (p ∨ q1) ∧ (p ∨ q2) ∧ ∙ ∙ ∙ ∧ (p
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