Introductory Statistics
3rd Edition
ISBN: 9780135188927
Author: Gould, Robert, Ryan, Colleen N. (colleen Nooter)
Publisher: Pearson,
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Chapter 4, Problem 87CRE
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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 4 Solutions
Introductory Statistics
Ch. 4 - GPA Predictors The scatterplots show SAT scores...Ch. 4 - Salary and Employment The first graph shows the...Ch. 4 - Age and Credits The scatterplot below shows data...Ch. 4 - Age and GPA The scatterplot shows data on age and...Ch. 4 - Credits and GPA (Example 1) The scatterplot shows...Ch. 4 - Salary and Education The scatterplot shows data on...Ch. 4 - Sisters and Brothers The scatterplot shows the...Ch. 4 - House Price and Area Describe the trend in the...Ch. 4 - Work and TV The scatterplot shows the number of...Ch. 4 - Work and Sleep The scatterplot shows the number of...
Ch. 4 - Age and Sleep The scatterplot shows the age and...Ch. 4 - Height and Weight for Women The figure shows a...Ch. 4 - College Tuition and ACT a. The first scatterplot...Ch. 4 - Ages of Women Who Give Birth The figure shows a...Ch. 4 - Law School The scatterplot shows the LSAT (Law...Ch. 4 - Medical School The scatterplot shows the...Ch. 4 - Matching Pick the letter of the graph that goes...Ch. 4 - Matching Pick the letter of the graph that goes...Ch. 4 - Matching Match each of the following correlations...Ch. 4 - Matching Match each of the following correlations...Ch. 4 - Airline Tickets (Example 2) The distance (in...Ch. 4 - Train Tickets The table for part (a) shows...Ch. 4 - SAT and GPA In Exercise 4.1 there is a graph of...Ch. 4 - House Price The correlation between house price...Ch. 4 - Rate My Professor Seth Wagerman, a former...Ch. 4 - Cousins Five people were asked how many female...Ch. 4 - GPA and Gym Use USA Today College published an...Ch. 4 - Education and Life Expectancy United Press...Ch. 4 - Salaries of College Graduates (Example 3) The...Ch. 4 - Mother and Daughter Heights The graph shows the...Ch. 4 - Median Weekly Earning by Gender The scatterplot...Ch. 4 - Home Prices and Size The scatterplot shows the...Ch. 4 - Height and Arm Span for Women (Example 4) TI-84...Ch. 4 - Hand and Foot Length for Women The computer output...Ch. 4 - Height and Arm Span for Men (Example 5)...Ch. 4 - Hand Length and Foot Length for Men Measurements...Ch. 4 - Comparing Correlation for Arm Span and Height The...Ch. 4 - Age and Weight for Men and Women The scatterplot...Ch. 4 - Singles and Doubles Winning Percentage The...Ch. 4 - Seesaw The figure shows a scatterplot of the...Ch. 4 - Choosing the Predictor and Response Variables...Ch. 4 - Choosing the Predictor and the Response Variable...Ch. 4 - Percentage of Smoke-Free Homes and Percentage of...Ch. 4 - Effect of Adult Smoking on High School Student...Ch. 4 - Car Insurance and Age The following graph shows...Ch. 4 - Life Insurance and Age The graph shows the monthly...Ch. 4 - How Is the Time of a Flight Related to the...Ch. 4 - Distance and Train Ticket Price The following...Ch. 4 - Do States with Higher Populations Have More...Ch. 4 - Movie Ratings The following table give the Rotten...Ch. 4 - Pitchers The table shows the Earned Run Average...Ch. 4 - Text Messages The following table shows the number...Ch. 4 - Answer the questions using complete sentences. a....Ch. 4 - Answer the questions using complete sentences. a....Ch. 4 - If there is a positive correlation between number...Ch. 4 - Suppose that the growth rate of children looks...Ch. 4 - Coefficient of Determination If the correlation...Ch. 4 - Coefficient of Determination Does a correlation of...Ch. 4 - Investing Some investors use a technique called...Ch. 4 - Blood Pressure Suppose a doctor telephones those...Ch. 4 - Salary and Year of Employment (Example 8) The...Ch. 4 - MPG: Highway and City The following figure shows...Ch. 4 - Cost of Turkeys The following table shows the...Ch. 4 - Wine Calories The table shows the calories in a...Ch. 4 - Teacher Pay and Expenditure Per Student (Example...Ch. 4 - Teacher Pay and High School Graduation Rates The...Ch. 4 - Does Having a Job Affect Students’ Grades?...Ch. 4 - Weight of Trash and Household Size Data were...Ch. 4 - Predicting Home Runs Data on the number of home...Ch. 4 - Predicting 3-Point Baskets Data on the 3-point...Ch. 4 - 4th-Grade Reading and Math Scores Data from the...Ch. 4 - SAT Scores The following table shows the average...Ch. 4 - Test Scores Assume that in a political science...Ch. 4 - Test Scores Assume that in a sociology class, the...Ch. 4 - Heights and Weights of People The following table...Ch. 4 - Heights and Weights of Men The table shows the...Ch. 4 - Prob. 77CRECh. 4 - Granola Bars The following table shows the fat...Ch. 4 - Shoe Size and Height The scatterplot shows the...Ch. 4 - Age and Sleep The following scatterplot shows the...Ch. 4 - Age and Weight The following scatterplot shows the...Ch. 4 - Heights and Test Scores a. The following figure...Ch. 4 - Law School Selectivity and Employment The...Ch. 4 - Prob. 84CRECh. 4 - Prob. 87CRECh. 4 - Fuel-Efficient Cars The following table gives the...Ch. 4 - Tall Buildings The following scatterplot shows...Ch. 4 - For 4.91 through 4.94 show your points in a rough...Ch. 4 - For 4.91 through 4.94 show your points in a rough...Ch. 4 - For 4.91 through 4.94 show your points in a rough...Ch. 4 - For 4.91 through 4.94 show your points in a rough...Ch. 4 - The following figure shows a scatterplot of the...Ch. 4 - Wages and Education The figure shows a scatterplot...Ch. 4 - Do Students Taking More Units Study More Hours?...Ch. 4 - Hours of Exercise and Hours of Homework The...Ch. 4 - Children’s Ages and Heights The following figure...Ch. 4 - Blackjack Tips The following figure shows the...Ch. 4 - Decrease in Cholesterol A doctor is studying...Ch. 4 - Prob. 102CRE
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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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