Mathematical Statistics with Applications
7th Edition
ISBN: 9781111798789
Author: Dennis O. Wackerly
Publisher: Cengage Learning
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Chapter 1, Problem 29SE
a.
To determine
Find the approximate fraction of the production of the machine that will fail to meet specifications.
b.
To determine
State the assumption regarding the diameter of the bearings that have been used in the previous part.
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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 1 Solutions
Mathematical Statistics with Applications
Ch. 1.1 - For each of the following situations, identify the...Ch. 1.2 - Are some cities more windy than others? Does...Ch. 1.2 - Of great importance to residents of central...Ch. 1.2 - The top 40 stocks on the over-the-counter (OTC)...Ch. 1.2 - Given here is the relative frequency histogram...Ch. 1.2 - The relative frequency histogram given next was...Ch. 1.2 - The self-reported heights of 105 students in a...Ch. 1.2 - An article in Archaeometry presented an analysis...Ch. 1.3 - Resting breathing rates for college-age students...Ch. 1.3 - It has been projected that the average and...
Ch. 1.3 - The following results on summations will help us...Ch. 1.3 - Prob. 12ECh. 1.3 - Prob. 13ECh. 1.3 - Refer to Exercise 1.3 and repeat parts (a) and (b)...Ch. 1.3 - Refer to Exercise 1.4 and repeat parts (a) and (b)...Ch. 1.3 - Prob. 16ECh. 1.3 - Prob. 17ECh. 1.3 - Prob. 18ECh. 1.3 - Prob. 19ECh. 1.3 - Prob. 20ECh. 1.3 - The manufacturer of a new food additive for beef...Ch. 1 - Prove that the sum of the deviations of a set of...Ch. 1 - The mean duration of television commercials is 75...Ch. 1 - Aqua running has been suggested as a method of...Ch. 1 - The following data give the lengths of time to...Ch. 1 - Prob. 26SECh. 1 - Prob. 27SECh. 1 - The discharge of suspended solids from a phosphate...Ch. 1 - Prob. 29SECh. 1 - Compared to their stay-at-home peers, women...Ch. 1 - Prob. 31SECh. 1 - Prob. 32SECh. 1 - Prob. 33SECh. 1 - Prob. 34SECh. 1 - Prob. 35SECh. 1 - Prob. 36SECh. 1 - Prob. 37SECh. 1 - Prob. 38SE
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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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