Statistics for Business & Economics, Revised (MindTap Course List)
12th Edition
ISBN: 9781285846323
Author: David R. Anderson, Dennis J. Sweeney, Thomas A. Williams, Jeffrey D. Camm, James J. Cochran
Publisher: South-Western College Pub
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Chapter 18.1, Problem 5E
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T1.4: Let ẞ(G) be the minimum size of a vertex cover, a(G) be the maximum size of an
independent set and m(G) = |E(G)|.
(i) Prove that if G is triangle free (no induced K3) then m(G) ≤ a(G)B(G). Hints - The
neighborhood of a vertex in a triangle free graph must be independent; all edges have at least
one end in a vertex cover.
(ii) Show that all graphs of order n ≥ 3 and size m> [n2/4] contain a triangle. Hints - you
may need to use either elementary calculus or the arithmetic-geometric mean inequality.
We consider the one-period model studied in class as an example. Namely, we assumethat the current stock price is S0 = 10. At time T, the stock has either moved up toSt = 12 (with probability p = 0.6) or down towards St = 8 (with probability 1−p = 0.4).We consider a call option on this stock with maturity T and strike price K = 10. Theinterest rate on the money market is zero.As in class, we assume that you, as a customer, are willing to buy the call option on100 shares of stock for $120. The investor, who sold you the option, can adopt one of thefollowing strategies: Strategy 1: (seen in class) Buy 50 shares of stock and borrow $380. Strategy 2: Buy 55 shares of stock and borrow $430. Strategy 3: Buy 60 shares of stock and borrow $480. Strategy 4: Buy 40 shares of stock and borrow $280.(a) For each of strategies 2-4, describe the value of the investor’s portfolio at time 0,and at time T for each possible movement of the stock.(b) For each of strategies 2-4, does the investor have…
Negate the following compound statement using De Morgans's laws.
Chapter 18 Solutions
Statistics for Business & Economics, Revised (MindTap Course List)
Ch. 18.1 - The following hypothesis test is to be conducted....Ch. 18.1 - Ten individuals participated in a taste test...Ch. 18.1 - The median number of part-time employees at...Ch. 18.1 - Net assets for the 50 largest stock mutual funds...Ch. 18.1 - The median annual income of subscribers to...Ch. 18.1 - The median annual income for families living in...Ch. 18.1 - Prob. 7ECh. 18.1 - A Pew Research Center survey asked adults if their...Ch. 18.1 - A poll taken during the recession in 2008 asked...Ch. 18.1 - Prob. 10E
Ch. 18.1 - Competition in the personal computer market is...Ch. 18.2 - Two fuel additives are tested to determine their...Ch. 18.2 - A sample of 10 men was used in a study to test the...Ch. 18.2 - Percents of on-time arrivals for flights in the...Ch. 18.2 - A test was conducted for two overnight mail...Ch. 18.2 - The PGA Players Championship was held at the...Ch. 18.2 - The Scholastic Aptitude Test (SAT) consists of...Ch. 18.3 - Two fuel additives are being tested to determine...Ch. 18.3 - Samples of starting annual salaries for...Ch. 18.3 - The gap between the earnings of men and women with...Ch. 18.3 - Prob. 21ECh. 18.3 - Each year Bloomberg Businessweek publishes...Ch. 18.3 - Police records show the following numbers of daily...Ch. 18.3 - A certain brand of microwave oven was priced at 10...Ch. 18.3 - Prob. 25ECh. 18.4 - A sample of 15 consumers provided the following...Ch. 18.4 - Three admission test preparation programs are...Ch. 18.4 - Forty-minute workouts of one of the following...Ch. 18.4 - Cond Nast Traveler magazine conducts an annual...Ch. 18.4 - A large corporation sends many of its first-level...Ch. 18.4 - The better-selling candies are often high in...Ch. 18.5 - Consider the following set of rankings for a...Ch. 18.5 - Prob. 33ECh. 18.5 - Prob. 34ECh. 18.5 - A national study by Harris Interactive, Inc.,...Ch. 18.5 - Prob. 36ECh. 18.5 - A student organization surveyed both current...Ch. 18 - A survey asked the following question: Do you...Ch. 18 - Due to a recent decline in the housing market, the...Ch. 18 - Twelve homemakers were asked to estimate the...Ch. 18 - Prob. 41SECh. 18 - The following data are product weights for the...Ch. 18 - A client wants to determine whether there is a...Ch. 18 - Prob. 44SECh. 18 - Prob. 45SECh. 18 - Prob. 46SECh. 18 - Prob. 47SE
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- Negate the following compound statement using De Morgans's laws.arrow_forwardQuestion 6: Negate the following compound statements, using De Morgan's laws. A) If Alberta was under water entirely then there should be no fossil of mammals.arrow_forwardNegate the following compound statement using De Morgans's laws.arrow_forward
- Characterize (with proof) all connected graphs that contain no even cycles in terms oftheir blocks.arrow_forwardLet G be a connected graph that does not have P4 or C3 as an induced subgraph (i.e.,G is P4, C3 free). Prove that G is a complete bipartite grapharrow_forwardProve sufficiency of the condition for a graph to be bipartite that is, prove that if G hasno odd cycles then G is bipartite as follows:Assume that the statement is false and that G is an edge minimal counterexample. That is, Gsatisfies the conditions and is not bipartite but G − e is bipartite for any edge e. (Note thatthis is essentially induction, just using different terminology.) What does minimality say aboutconnectivity of G? Can G − e be disconnected? Explain why if there is an edge between twovertices in the same part of a bipartition of G − e then there is an odd cyclearrow_forward
- Let G be a connected graph that does not have P4 or C4 as an induced subgraph (i.e.,G is P4, C4 free). Prove that G has a vertex adjacent to all othersarrow_forwardWe consider a one-period market with the following properties: the current stock priceis S0 = 4. At time T = 1 year, the stock has either moved up to S1 = 8 (with probability0.7) or down towards S1 = 2 (with probability 0.3). We consider a call option on thisstock with maturity T = 1 and strike price K = 5. The interest rate on the money marketis 25% yearly.(a) Find the replicating portfolio (φ, ψ) corresponding to this call option.(b) Find the risk-neutral (no-arbitrage) price of this call option.(c) We now consider a put option with maturity T = 1 and strike price K = 3 onthe same market. Find the risk-neutral price of this put option. Reminder: A putoption gives you the right to sell the stock for the strike price K.1(d) An investor with initial capital X0 = 0 wants to invest on this market. He buysα shares of the stock (or sells them if α is negative) and buys β call options (orsells them is β is negative). He invests the cash balance on the money market (orborrows if the amount is…arrow_forwardDetermine if the two statements are equivalent using a truth tablearrow_forward
- Question 4: Determine if pair of statements A and B are equivalent or not, using truth table. A. (~qp)^~q в. р л~9arrow_forwardDetermine if the two statements are equalivalent using a truth tablearrow_forwardQuestion 3: p and q represent the following simple statements. p: Calgary is the capital of Alberta. A) Determine the value of each simple statement p and q. B) Then, without truth table, determine the va q: Alberta is a province of Canada. for each following compound statement below. pvq р^~q ~рл~q ~q→ p ~P~q Pq b~ (d~ ← b~) d~ (b~ v d) 0 4arrow_forward
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