...CHAPTER 6: THE NORMAL DISTRIBUTION AND OTHER CONTINUOUS DISTRIBUTIONS 1. In its standardized form, the normal distribution a) has a mean of 0 and a standard deviation of 1. b) has a mean of 1 and a variance of 0. c) has an area equal to 0.5. d) cannot be used to approximate discrete probability distributions. ANSWER: a TYPE: MC DIFFICULTY: Easy KEYWORDS: standardized normal distribution, properties 2. Which of the following about the normal distribution is NOT true? a) Theoretically, the mean, median, and mode are the same. b) About 2/3 of the observations fall within 1 standard deviation from the mean. c) It is a discrete probability distribution. d) Its parameters are the mean, , and standard deviation, . ANSWER: c TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 3. If a particular batch of data is approximately normally distributed, we would find that approximately a) 2 of every 3 observations would fall between 1 standard deviation around the mean. b) 4 of every 5 observations would fall between 1.28 standard deviations around the mean. c) 19 of every 20 observations would fall between 2 standard deviations around the mean. d) all of the above ANSWER: d TYPE: MC DIFFICULTY: Easy KEYWORDS: normal distribution, properties 4. For some positive value of Z, the probability that a standardized normal variable is between 0 and Z is 0.3770. The value of Z is a) 0.18. b) 0.81. c) 1.16. d) 1...
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...questions. Use only blanks left to answer lab questions. Provide all histograms you are asked to print, but DO NOT print data you are asked to generate. 1. Continuous distributions: Generate and store in column c1 10,000 values from the uniform distribution on the interval [3,7] as follows: random 10000 c1; uniform 3 7. [3] a. Use mean command to find the sample mean x of these data———————– ¯ [2] b. What is the mean µ of the uniform distribution on the interval [3,7]?————[1] c. Compare µ to the value x you found in part a). ———————– ¯ Generate and store in column c2 1,000 values from exponential distribution with parameter λ = .125 as follows: random 1000 c2; exponential 8. Note: The mean µ and the standard deviation σ of such distribution are both equal to 1/λ = 8 and this is the value you are asked to enter in the command above. [3] d. Use desc command to find the sample mean x and sample standard deviation s for ¯ these 1,000 data —————– and —————— Are x and s close to the value 1/λ = 8?———————– Why?——————————¯ [3] e. Print (and include in your assignment) the histogram of the 1,000 values you generated from this exponential distribution. What is the shape of this distribution?———————– 2. Normal distribution: Generate and store in column c3 10,000 values from the standard normal distribution as follows: random 10000 c3; normal. [3] a. Print (and include in your assignment) the histogram for these data. What is the shape of this histogram?———————————– [3] b. What is the value...
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...taken from past examinations. The weight of topics in these sample questions is not representative of the weight of topics on the exam. The syllabus indicates the exam weights by topic. Copyright 2013 by the Society of Actuaries and the Casualty Actuarial Society C-09-08 PRINTED IN U.S.A. 1. You are given: (i) Losses follow a loglogistic distribution with cumulative distribution function: bx / θ g F b xg = 1+ bx / θ g γ γ (ii) The sample of losses is: 10 35 80 86 90 120 158 180 200 210 1500 Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile estimates. (A) (B) (C) (D) (E) Less than 77 At least 77, but less than 87 At least 87, but less than 97 At least 97, but less than 107 At least 107 2. You are given: (i) The number of claims has a Poisson distribution. (ii) (iii) (iv) Claim sizes have a Pareto distribution with parameters θ = 0.5 and α = 6 . The number of claims and claim sizes are independent. The observed pure premium should be within 2% of the expected pure premium 90% of the time. Determine the expected number of claims needed for full credibility. (A) (B) (C) (D) (E) Less than 7,000 At least 7,000, but...
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...A FIRST COURSE IN PROBABILITY This page intentionally left blank A FIRST COURSE IN PROBABILITY Eighth Edition Sheldon Ross University of Southern California Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. A first course in probability / Sheldon Ross. — 8th ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X 1. Probabilities—Textbooks. I. Title. QA273.R83 2010 519.2—dc22 2008033720 Editor in Chief, Mathematics and Statistics: Deirdre Lynch Senior Project Editor: Rachel S. Reeve Assistant Editor: Christina Lepre Editorial Assistant: Dana Jones Project Manager: Robert S. Merenoff Associate Managing Editor: Bayani Mendoza de Leon Senior Managing Editor: Linda Mihatov Behrens Senior Operations Supervisor: Diane Peirano Marketing Assistant: Kathleen DeChavez Creative Director: Jayne Conte Art Director/Designer: Bruce Kenselaar AV Project Manager: Thomas Benfatti Compositor: Integra Software Services Pvt. Ltd, Pondicherry, India Cover Image Credit: Getty Images, Inc. © 2010, 2006, 2002, 1998, 1994, 1988, 1984, 1976 by Pearson Education, Inc., Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc...
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...Jankovi´, Ivan Mati´, and Nikola Petrovi´ s c c c c Problem-Solving Strategies by Arthur Engel Problems in Analysis by Bernard R. Gelbaum Problems in Real and Complex Analysis by Bernard R. Gelbaum (continued after subject index) Wolfgang Schwarz 40 Puzzles and Problems in Probability and Mathematical Statistics Wolfgang Schwarz Universit¨ t Potsdam a Humanwissenschaftliche Fakult¨ t a Karl-Liebknecht Strasse 24/25 D-14476 Potsdam-Golm Germany wschwarz@uni-potsdam.de Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA Peter.winkler@dartmouth.edu ISBN-13: 978-0-387-73511-5 e-ISBN-13: 978-0-387-73512-2 Mathematics Subject Classification (2000): 60-xx Library of Congress Control Number: 2007936604 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts...
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...researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research. http://www.jstor.org Perishable Inventory Theory: A Review STEVEN NAHMIAS The University of Santa Clara, Santa Clara, California (ReceivedFebruary1980;acceptedJanuary1982) This paper reviews the relevant literature on the problem of determining suitable ordering policies for both fixed life perishable inventory, and inventory subject to continuous exponential decay. We consider both deterministic and stochastic demand for...
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...Erlang Distribution If interarrival or service times are not exponential, an Erlang random variable can often be used to model them. If T is an Erlang random variable with rate parameter R and shape parameter k, the density of T is given by R(Rt)kϪ1eϪRt f (t) ϭ ᎏᎏ (k Ϫ 1)! (t Ն 0) and k E(T) ϭ ᎏᎏ R and k var T ϭ ᎏᎏ R2 Birth–Death Processes For a birth-death process, the steady-state probability (pj) or fraction of the time that the process spends in state j can be found from the following flow balance equations: ( j ϭ 0) ( j ϭ 1) ( j ϭ 2) и и и ( jth equation) p0l0 ϭ p1m1 (l1 ϩ m1)p1 ϭ l0p0 ϩ m2p2 (l2 ϩ m2)p2 ϭ l1p1 ϩ m3p3 (lj ϩ mj)pj ϭ ljϪ1pjϪ1 ϩ mjϩ1pjϩ1 The jth flow balance equation states that the expected number of transitions per unit time out of state j ϭ (expected number of transitions per unit time into state j). The solution to the balance equations is found from l0l1 и и и ljϪ1 pj ϭ p0 ᎏᎏ m1m2 и и и mj ( j ϭ 1, 2, . . .) and the fact that p0 ϩ p1 ϩ и и и ϭ 1. Notation for Characteristics of Queuing Systems pj L Lq Ls W Wq Ws l m ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ steady-state probability that j customers are in system expected number of customers in system expected number of customers in line (queue) expected number of customers in service expected time a customer spends in system expected time a customer spends waiting in line expected time a customer spends in service average number of customers...
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..."what-if" questions. * Does not interfere with real-world systems. * Study interactions among variable. * "Time comparison" is possible. * Handles complications that other methods can't. Disadvantages of Simulation: * Can be expensive and time consuming. * Does not generate optimal solutions. * Managers must generate all conditions and constraints. * Each model is unique. Monte Carlo Simulation: Can be used with variables that are probabilistic. Steps: * Establish the probability distribution for each random variable. * Use random numbers to generate random values. * Repeat for some number of replications. Probability Distributions: Historical data Goodness-of-fit tests for common distributions: * Normal * Uniform * Exponential * Poisson * Binomial Role of Computers: * Built-in random number procedures for simulating from several different probability distributions. * Easy and fast replication of a model. * Change input values and track output measures. Simulation Software: * General purpose languages * Visual Basic, C++, Fortan, etc....
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...When the EB method is used in complex models with many parameters, it often requires the computation of multidimensional definite integrals. This requirement has effectively placed practical limits on the complexity of the problems a long time in the past. However, the parallel development of a Monte Carlo simulation based numerical strategies have revolutionized the applied Bayesian data analysis with the availability of super computing power over the past number of decades. These strategies provide a structure within which many complex Bayesian models can be analyzed using generic software. Moreover, the simulation study is a tractable practice that can be utilized to examine and compare the performance of different estimators empirically. In this study, one generates a sample of random data in such a way that mimics a real problem and recapitulates...
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...1. 2. 3. 4. 5. System Reliability function Probability distributions Reliability Block Diagram Serial and Parallel Structures Stand-by Structure k-out-of n Structure Complex structure 2. Part 2: Reliability of Structures 3. Part 3: Reliability Allocation 4. References 2 2 2 2 2 2 2 2 Chapter 3 - Part 1: Basic Reliability Models SYSTEM System: a collection of components or items performing a specific function. 4 4 4 4 4 4 4 4 STATE OF A SYSTEM A system is considered to be in one of the two following states: In operation (Up) Failed (Down) Transition from one state to the other occurs according to a known or unknown probability function. de aF de aF delllliiiiaF de aF 5 5 5 5 5 5 5 5 noitca ecnanetniaM noitca ecnanetniaM noitca ecnanetniaM noitca ecnanetniaM riapeR riapeR riapeR riapeR e eruliaF gn arepO gn arepO gniiiittttarepO gn arepO SYSTEM LIFETIME Lifetime is a measure of performance. Lifetime is a measure of performance. In general, lifetime is measured by the number In general, lifetime is measured by the of hours the system was in operation. number of hours the system was in operation. It can also be measured by the number of It can also be measured by the number of km or miles raced, number of pages copied, pages copied, km or miles raced, number wheel rotations, …etc. of wheel rotations, …etc. Lifetime (T) is a nonnegative random variable. Lifetime (T) is a nonnegative random variable. 6 6 6 6 6 6 6 6 LIFETIME PROBABILITY DENSITY...
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...Starbucks Queuing Model Inevitably, for those coffee drinkers that don’t have a coffee machine at home, it is a fact of life that you would have to wait in lines at your closest coffee shop to get that glorious cup of Joe. For the one of the most prolific coffee chains, Starbucks, this is a problem that needs to be addressed as customer service is one of the cornerstones of the industry. In order maintain a good customer service, Starbucks must minimize the in-line wait time and the service times so that customers can go in and out of the shop at their own leisure. Faster service means more customers can be helped in a short amount of time as well as keeping lines short during rush-hour times. Shorter lines, in turn, means that as customers enters the store they will be less likely to be discouraged by the amount of waiting they would have to do and leave for other source. The best way that this problem may be resolved is through analyzing and simulating the queuing behavior of customers as they enter store to make their purchases. Through observations and statistical data, we seek to utilize a model to simulate this behavior as well as determine what variables are critical to minimizing customer in line waiting times as well as customer service times. Background Starbucks was formed in Seattle, Washington back in 1971 by three partners. They originally were not a coffee shop. Rather, they were a provider of whole coffee beans to restaurants and other coffee bars. It was not...
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...Probability is a numerical measure of the likelihood that an event will occur. Probability values are always assigned on a scale from 0 to 1…..A probability near zero indicates an event is quite unlikely to occur ..A probability near one indicates an event is almost certain to occur. An experiment is any process that generates well- defined outcomes. The sample space for an experiment is the set of all experimental outcomes.. An experimental outcome is also called a sampl point. The probability assigned to each experimental outcome must be between 0 and 1, inclusively Classical Method: Assigning probabilities based on the assumption of equally likely outcomes… Relative Frequency Method: Assigning probabilities based on experimentation or historical data ….Subjective Method; Assigning probabilities based on judgment…….. An event is a collection of sample points. …..The probability of any event is equal to the sum of the probabilities of the sample points in the event If we can identify all the sample points of an experiment and assign a probability to each, we can compute the probability of an event ….There are some basic probability relationships that can be used to compute the probability of an event without knowledge of all the sample point probabilities.( Complement of an Event,, Union of Two Events.. Intersection of Two Events.. Mutually Exclusive Events) …..The complement of event A is defined to be the eventconsisting of all sample points that are not in A....
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...quartile is the number that spits a set of data into quarters; thus, there is the first quartile, the second (median) and the third quartile (Yakir, 2011, p.g 32). b) Write a brief definition of the word "quantile" as it might be used in statistics. Be sure to provide a citation (do not cut and paste... use your own words to summarize what you discovered): A quantile is refers to aa set of values that split an ordered set of data, into equal parts. For instance, a quartile is a...
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...Distributions, Sampling and Estimation Our main objective in this section is to learn how to estimate some important characteristics of random variables using data. We will concentrate on estimating the mean from a normal distribution, and the proportion of success in a binomial distribution. To talk about estimation, we first need to know something about distributions and sampling. Most data used for decision making exhibit variation. We are interested in drawing conclusions from such data. Probability and statistics give us the necessary tools. We will again make use of the concept of a random variable. We have reviewed discrete random variables already. Before moving on, we need to review continuous random variables. Continuous random variables take on continuous or interval values (there are an infinite number of possibilities). If you are measuring, the distribution of the result will almost always be continuous. For example, the width of an extruded bar is a continuous random variable. The distribution of a continuous random variable is represented by a continuous curve (called the probability density function (pdf) and often denoted f(y)). The height of the curve does not represent probabilities; instead, the area under the curve between two points tells us about the probability. As a result, the probability that a continuous random variable is exactly equal to a single value is 0 (there is no area under a single...
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...be measured over an extended period of time. • We assume arrival times and service times are random. • • • • Assumptions Independent arrivals Exponential distributions Customers do not leave or change queues. Large queues do not discourage customers. Many assumptions are not always true, but queuing theory gives good results anyway Queuing Model Q W λ Tw Tq S Interesting Values • Arrival rate (λ) — the average rate at which customers arrive. • Service time (s) — the average time required to service one customer. • Number waiting (W) — the average number of customers waiting. • Number in the system (Q) — the average total number of customers in the system. More Interesting Values • Time in the system (Tq) the average time each customer is in the system, both waiting and being serviced. Time waiting (Tw) the average time each customer waits in the queue. Tq = Tw + s Arrival Rate • The arrival rate, λ, is the average rate new customers arrive measured in arrivals per time period. Common units are access/second • The inter-arrival time, a, is the average time between customer arrivals. It is measured in time per customer. A common unit would be seconds/access. • a=1/λ Random Values • We assume that most of the events we are interested in occur randomly. – Time of a request to a device – Time to service a request – Time user makes a request Exponential...
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