Applied Statistics and Probability for Engineers
6th Edition
ISBN: 9781118539712
Author: Douglas C. Montgomery
Publisher: WILEY
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Chapter 2.4, Problem 99E
a.
To determine
Find the
b.
To determine
Find the probability of B.
c.
To determine
Find the probability of
d.
To determine
Find the probability of
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Does there appear to be a positive or negative relationship between price and screen size? Use a scatter plot to examine the relationship. How to take snapshots: if you use a MacBook, press Command+ Shift+4 to take snapshots. If you are using Windows, use the Snipping Tool to take snapshots.
Question 1: Determine and interpret the correlation coefficient between the two variables. In your interpretation, discuss the direction of the relationship (positive, negative, or zero relationship). Also discuss the strength of the relationship.
Value of correlation coefficient:
Direction of the relationship (positive, negative, or zero relationship):
Strength of the relationship (strong/moderate/weak):
Question 2: Estimate the relationship between screen size and price using a simple linear regression model and interpret the estimated coefficients. In your interpretation, tell the dollar amount by which price will change for each unit of increase in screen size. (The answer for the…
In this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0
given (under the measure P) by
d.St 0.03 St dt + 0.2 St dwt,
with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to
price an option on this stock (which we name cubic put). This option is European-type, with
maturity 3 months (i.e. T = 0.25 years), and payoff given by
F = (8-5)+
(a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure
Q. (You don't need to prove it, simply give the answer.)
(b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2.
(c) Let X =
S. Find the Stochastic Differential Equation satisfied by the process (Xt)
under the measure Q.
(d) Find an explicit expression for X₁ = S3 under measure Q.
(e) Using the results above, find the price of the cubic put option mentioned above.
(f) Is the price in (e) the same as in question (b)? (Explain why.)
Problem 4. Margrabe formula and the Greeks (20 pts)
In the homework, we determined the Margrabe formula for the price of an option allowing you to
swap an x-stock for a y-stock at time T. For stocks with initial values xo, yo, common volatility
σ and correlation p, the formula was given by
Fo=yo (d+)-x0Þ(d_),
where
In (±²
Ꭲ
d+
õ√T
and
σ = σ√√√2(1 - p).
дго
(a) We want to determine a "Greek" for ỡ on the option: find a formula for
θα
(b) Is
дго
θα
positive or negative?
(c) We consider a situation in which the correlation p between the two stocks increases: what
can you say about the price Fo?
(d) Assume that yo< xo and p = 1. What is the price of the option?
Chapter 2 Solutions
Applied Statistics and Probability for Engineers
Ch. 2.1 - Prob. 1ECh. 2.1 - Prob. 2ECh. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...
Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - Provide a reasonable description of the sample...Ch. 2.1 - 2-18. Three attempts are made to read data in a...Ch. 2.1 - 2-19. Three events are shown on the Venn diagram...Ch. 2.1 - 2-20. Three events are shown on the Venn diagram...Ch. 2.1 - 2-21. A digital scale that provides weights to the...Ch. 2.1 - Prob. 22ECh. 2.1 - 2-23. Go Tutorial Four bits are transmitted over...Ch. 2.1 - 2-24. In light-dependent photosynthesis, light...Ch. 2.1 - 2-25. In control replication, cells are replicated...Ch. 2.1 - 2-26. Disks of polycarbonate plastic from a...Ch. 2.1 - 2-27. Samples of a cast aluminum part are...Ch. 2.1 - 2-28. Samples of emissions from three suppliers...Ch. 2.1 - 2-29. The rise time of a reactor is measured in...Ch. 2.1 - Prob. 30ECh. 2.1 - Prob. 31ECh. 2.1 - 2-32. Counts of the Web pages provided by each of...Ch. 2.1 - 2-33. A reactor’s rise time is measured in minutes...Ch. 2.1 - 2-34. A wireless garage door opener has a code...Ch. 2.1 - Prob. 35ECh. 2.1 - 2-36. In a manufacturing operation, a part is...Ch. 2.1 - 2-37. New designs for a wastewater treatment tank...Ch. 2.1 - 2-38. A manufacturing process consists of 10...Ch. 2.1 - Prob. 39ECh. 2.1 - 2-40. In a sheet metal operation, three notches...Ch. 2.1 - Prob. 41ECh. 2.1 - 2-42. In the layout of a printed circuit board for...Ch. 2.1 - 2-43. In the laboratory analysis of samples from a...Ch. 2.1 - 2-44. In the design of an electromechanical...Ch. 2.1 - 2-45. Consider the design of a communication...Ch. 2.1 - 2-46. A byte is a sequence of eight bits and each...Ch. 2.1 - 2-47. In a chemical plant, 24 holding tanks are...Ch. 2.1 - 2-48. Plastic parts produced by an...Ch. 2.1 - 2-49. A bin of 50 parts contains 5 that are...Ch. 2.1 - 2-50. The following table summarizes 204...Ch. 2.1 - 2-51. A Web ad can be designed from four different...Ch. 2.1 - 2-52. Consider the hospital emergency department...Ch. 2.1 - 2-53. An article in The Journal of Data Science...Ch. 2.1 - 2-54. Similar to the hospital schedule in Example...Ch. 2.1 - 2-55. Consider the bar code in Example 2-12. One...Ch. 2.1 - 2-56. A computer system uses passwords that...Ch. 2.1 - Prob. 57ECh. 2.2 - 2-58. Each of the possible five outcomes of a...Ch. 2.2 - 2-59. The sample space of a random experiment is...Ch. 2.2 - 2-60. Orders for a computer are summarized by the...Ch. 2.2 - Prob. 61ECh. 2.2 - Prob. 62ECh. 2.2 - 2-63. An injection-molded part is equally likely...Ch. 2.2 - 2-64. In an acid-base titration, a base or acid is...Ch. 2.2 - Prob. 65ECh. 2.2 - Prob. 66ECh. 2.2 - 2-67. Suppose your vehicle is licensed in a state...Ch. 2.2 - Prob. 68ECh. 2.2 - 2-69. Magnesium alkyls are used as homogenous...Ch. 2.2 - 2-70. Disks of polycarbonate plastic from a...Ch. 2.2 - 2-71. Samples of emissions from three suppliers...Ch. 2.2 - Prob. 72ECh. 2.2 - Prob. 73ECh. 2.2 - Prob. 74ECh. 2.2 - 2-75. A Web ad can be designed from four different...Ch. 2.2 - Prob. 76ECh. 2.2 - Prob. 77ECh. 2.2 - 2-78. Consider the bar code in Example 2-12....Ch. 2.2 - 2-79. Similar to the hospital schedule in Example...Ch. 2.2 - Prob. 80ECh. 2.2 - 2-81. A computer system uses passwords that...Ch. 2.3 - 2-82. If P(A) = 0.3, P(B) = 0.2, and P(A B) =...Ch. 2.3 - 2-83. If A, B, and C are mutually exclusive events...Ch. 2.3 - 2-84. In the article “ACL Reconstruction Using...Ch. 2.3 - 2-85. Disks of polycarbonate plastic from a...Ch. 2.3 - 2-86. Strands of copper wire from a manufacturer...Ch. 2.3 - 2-87. The analysis of shafts for a compressor is...Ch. 2.3 - 2-88. Cooking oil is produced in two main...Ch. 2.3 - 2-89. A manufacturer of front lights for...Ch. 2.3 - 2-90. A computer system uses passwords that are...Ch. 2.3 - 2-91. Consider the endothermic reactions in...Ch. 2.3 - 2-92. A Web ad can be designed from four different...Ch. 2.3 - Prob. 93ECh. 2.3 - Prob. 94ECh. 2.3 - 2-95. Consider the barcode in Example 2-12....Ch. 2.3 - Prob. 96ECh. 2.3 - Prob. 97ECh. 2.3 - Prob. 98ECh. 2.4 - 2-99. Disks of polycarbonate plastic from a...Ch. 2.4 - 2-100. Samples of skin experiencing desquamation...Ch. 2.4 - 2-101. The analysis of results from a leaf...Ch. 2.4 - 2-102. Samples of a cast aluminum part are...Ch. 2.4 - 2-103. The following table summarizes the analysis...Ch. 2.4 - 2-104. Consider the data on wafer contamination...Ch. 2.4 - 2-105. The following table summarizes the number...Ch. 2.4 - 2-106. A maintenance firm has gathered the...Ch. 2.4 - 2-107. A lot of 100 semiconductor chips contains...Ch. 2.4 - 2-108. A batch of 500 containers for frozen orange...Ch. 2.4 - 2-109. A batch of 350 samples of rejuvenated...Ch. 2.4 - 2-110. A computer system uses passwords that are...Ch. 2.4 - 2-111. If , must A = B? Draw a Venn diagram to...Ch. 2.4 - 2-112. Suppose A and B are mutually exclusive...Ch. 2.4 - 2-113. Consider the endothermic reactions in...Ch. 2.4 - Prob. 114ECh. 2.4 - Prob. 115ECh. 2.4 - 2-116. An article in the The Canadian Entomologist...Ch. 2.4 - 2-117. Consider the bar code in Example 2-12....Ch. 2.4 - Prob. 118ECh. 2.4 - Prob. 119ECh. 2.4 - 2-120. A computer system uses passwords that...Ch. 2.5 - 2-121. Suppose that and . Determine the...Ch. 2.5 - Prob. 122ECh. 2.5 - 2-123. The probability is 1% that an electrical...Ch. 2.5 - 2-124. Suppose 2% of cotton fabric rolls and 3% of...Ch. 2.5 - 2-125. The edge roughness of slit paper products...Ch. 2.5 - 2-126. In the 2012 presidential election, exit...Ch. 2.5 - 2-127. Computer keyboard failures are due to...Ch. 2.5 - 2-128. Heart failures are due to either natural...Ch. 2.5 - Prob. 129ECh. 2.5 - Prob. 130ECh. 2.5 - Prob. 131ECh. 2.5 - 2-132. Consider the endothermic reactions in...Ch. 2.5 - Prob. 133ECh. 2.5 - 2-134. Consider the hospital emergency room data...Ch. 2.5 - Prob. 135ECh. 2.5 - Prob. 136ECh. 2.5 - 2-137. A Web ad can be designed from four...Ch. 2.5 - 2-138. Consider the code in Example 2-12. Suppose...Ch. 2.5 - 2-139. Similar to the hospital schedule in Example...Ch. 2.5 - Prob. 140ECh. 2.5 - 2-141. A computer system uses passwords that...Ch. 2.6 - 2-142. If , and , are the events A and B...Ch. 2.6 - 2-143. If , and , are the events B and the...Ch. 2.6 - 2-144. If , and A and B are mutually exclusive,...Ch. 2.6 - 2-145. A batch of 500 containers of frozen orange...Ch. 2.6 - 2-146. Disks of polycarbonate plastic from a...Ch. 2.6 - 2-147. Samples of emissions from three suppliers...Ch. 2.6 - 2-148. Redundant array of inexpensive disks (RAID)...Ch. 2.6 - 2-149. The probability that a lab specimen...Ch. 2.6 - 2-150. In a test of a printed circuit board using...Ch. 2.6 - 2-151. Six tissues are extracted from an ivy plant...Ch. 2.6 - 2-152. A player of a video game is confronted with...Ch. 2.6 - 2-153. In an acid-base titration, a base or acid...Ch. 2.6 - Prob. 154ECh. 2.6 - Prob. 155ECh. 2.6 - 2-156. The following circuit operates if and only...Ch. 2.6 - Prob. 157ECh. 2.6 - Prob. 158ECh. 2.6 - Prob. 159ECh. 2.6 - Prob. 160ECh. 2.6 - 2-161. A Web ad can be designed from four...Ch. 2.6 - Prob. 162ECh. 2.6 - Prob. 163ECh. 2.6 - 2-164. Table 2-1 provides data on wafers...Ch. 2.6 - 2-165. Table 2-1 provides data on wafers...Ch. 2.7 - Prob. 166ECh. 2.7 - Prob. 167ECh. 2.7 - Prob. 168ECh. 2.7 - Prob. 169ECh. 2.7 - 2-170. In the 2012 presidential election, exit...Ch. 2.7 - Prob. 171ECh. 2.7 - 2-172. An inspector working for a manufacturing...Ch. 2.7 - 2-173. A new analytical method to detect...Ch. 2.7 - 2-174. Consider the endothermic reactions in...Ch. 2.7 - Prob. 175ECh. 2.7 - Prob. 176ECh. 2.7 - 2-177. Two Web colors are used for a site...Ch. 2.7 - Prob. 178ECh. 2.7 - 2-179. An e-mail filter is planned to separate...Ch. 2.7 - 2-180. A recreational equipment supplier finds...Ch. 2.7 - Prob. 181ECh. 2.8 - 2-182. Decide whether a discrete or continuous...Ch. 2.8 - 2-183. Decide whether a discrete or continuous...Ch. 2.8 - 2-184. Decide whether a discrete or continuous...Ch. 2 - Prob. 185SECh. 2 - 2-186. A sample of three calculators is selected...Ch. 2 - 2-187. Samples of a cast aluminum part are...Ch. 2 - 2-188. Shafts are classified in terms of the...Ch. 2 - 2-189. If A, B, and C are mutually exclusive...Ch. 2 - 2-190. The analysis of shafts for a compressor is...Ch. 2 - 2-191. A researcher receives 100 containers of...Ch. 2 - 2-192. A lot contains 15 castings from a local...Ch. 2 - 2-193. In the manufacturing of a chemical...Ch. 2 - 2-194. Incoming calls to a customer service...Ch. 2 - 2-195. A congested computer network has a 0.002...Ch. 2 - 2-196. Samples of a cast aluminum part are...Ch. 2 - 2-197. Go Tutorial An optical storage device uses...Ch. 2 - 2-198. Semiconductor lasers used in optical...Ch. 2 - 2-199. Energy released from cells breaks the...Ch. 2 - 2-200. A sample preparation for a chemical...Ch. 2 - 2-201. In circuit testing of printed circuit...Ch. 2 - 2-202. The data from 200 machined parts are...Ch. 2 - 2-203. Computers in a shipment of 100 units...Ch. 2 - 2-204. The probability that a customer’s order is...Ch. 2 - 2-205. Let E1, E2 and E3 denote the samples that...Ch. 2 - 2-206. Transactions to a computer database are...Ch. 2 - Prob. 207SECh. 2 - Prob. 208SECh. 2 - Prob. 209SECh. 2 - 2-210. The British government has stepped up its...Ch. 2 - Prob. 211SECh. 2 - 2-212. An encryption-decryption system consists of...Ch. 2 - 2-213. It is known that two defective copies of a...Ch. 2 - 2-214. A robotic insertion tool contains 10...Ch. 2 - Prob. 215SECh. 2 - Prob. 216SECh. 2 - 2-217. A lot of 50 spacing washers contains 30...Ch. 2 - Prob. 218SECh. 2 - Prob. 219SECh. 2 - Prob. 220SECh. 2 - 2-221. The following circuit operates if and only...Ch. 2 - 2-222. A company that tracks the use of its Web...Ch. 2 - Prob. 223SECh. 2 - Prob. 224SECh. 2 - Prob. 225SECh. 2 - 2-226. Natural red hair consists of two genes....Ch. 2 - Prob. 227SECh. 2 - Prob. 228SECh. 2 - Prob. 229SECh. 2 - Prob. 230SECh. 2 - Prob. 231SECh. 2 - 2-232. A biotechnology manufacturing firm can...Ch. 2 - 2-233. Assume the following characteristics of the...Ch. 2 - Prob. 234SECh. 2 - Prob. 235SE
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- We consider a 4-dimensional stock price model given (under P) by dẴ₁ = µ· Xt dt + йt · ΣdŴt where (W) is an n-dimensional Brownian motion, π = (0.02, 0.01, -0.02, 0.05), 0.2 0 0 0 0.3 0.4 0 0 Σ= -0.1 -4a За 0 0.2 0.4 -0.1 0.2) and a E R. We assume that ☑0 = (1, 1, 1, 1) and that the interest rate on the market is r = 0.02. (a) Give a condition on a that would make stock #3 be the one with largest volatility. (b) Find the diversification coefficient for this portfolio as a function of a. (c) Determine the maximum diversification coefficient d that you could reach by varying the value of a? 2arrow_forwardQuestion 1. Your manager asks you to explain why the Black-Scholes model may be inappro- priate for pricing options in practice. Give one reason that would substantiate this claim? Question 2. We consider stock #1 and stock #2 in the model of Problem 2. Your manager asks you to pick only one of them to invest in based on the model provided. Which one do you choose and why ? Question 3. Let (St) to be an asset modeled by the Black-Scholes SDE. Let Ft be the price at time t of a European put with maturity T and strike price K. Then, the discounted option price process (ert Ft) t20 is a martingale. True or False? (Explain your answer.) Question 4. You are considering pricing an American put option using a Black-Scholes model for the underlying stock. An explicit formula for the price doesn't exist. In just a few words (no more than 2 sentences), explain how you would proceed to price it. Question 5. We model a short rate with a Ho-Lee model drt = ln(1+t) dt +2dWt. Then the interest rate…arrow_forwardIn this problem, we consider a Brownian motion (W+) t≥0. We consider a stock model (St)t>0 given (under the measure P) by d.St 0.03 St dt + 0.2 St dwt, with So 2. We assume that the interest rate is r = 0.06. The purpose of this problem is to price an option on this stock (which we name cubic put). This option is European-type, with maturity 3 months (i.e. T = 0.25 years), and payoff given by F = (8-5)+ (a) Write the Stochastic Differential Equation satisfied by (St) under the risk-neutral measure Q. (You don't need to prove it, simply give the answer.) (b) Give the price of a regular European put on (St) with maturity 3 months and strike K = 2. (c) Let X = S. Find the Stochastic Differential Equation satisfied by the process (Xt) under the measure Q. (d) Find an explicit expression for X₁ = S3 under measure Q. (e) Using the results above, find the price of the cubic put option mentioned above. (f) Is the price in (e) the same as in question (b)? (Explain why.)arrow_forward
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