Lecture 6. Poisson processes
|
|
- Kristin O’Connor’
- 7 years ago
- Views:
Transcription
1 Lecture 6. Poisson processes Mathematical Statistics and Discrete Mathematics November 18th, / 15
2 Motivation We want to have a probabilistic model describing the number of calls arriving at a call center within a given time range, say, 15 minutes between 1pm and 1.15pm. We can represent the incoming calls as marks on the timeline: So that it is a realistic model, it should posses the following properties: The numbers of calls arriving in disjoint time intervals should be independent The distribution of the number of calls arriving in a certain time interval depends only on the length of the interval and not its placement in time We will define a stochastic process which satisfies these properties, namely the Poisson process. 2 / 15
3 Poisson distribution A discrete random variable X has Poisson distribution with parameter λ > 0 if f X (k) = P(X = k) = λk k! e λ, for k = 0, 1, 2,... We denote this by writing X Pois(λ). Properties: E[X] = Var[X] = λ. If X Pois(λ 1 ), Y Pois(λ 2 ), and X and Y are independent, then X + Y Pois(λ 1 + λ 2 ). Pois(λ) can be well approximated by Binom(n, p) with very large n and small p such that np = λ. Hence, it models well the number of successes in a large number of trials where the success probability is small. This is sometimes called the law of rare events. 3 / 15
4 Poisson distribution Poisson distribution can be used to model the number of people winning the lottery each week, people older than 100 years in some community, deaths in a given age group within a year, mutations in a piece of DNA after exposure to radiation, times a server is accessed per minute, telephone calls arriving at a call center within 15 minutes (the population covered by one call center is large and each person has a small probability of making a call within a particular 15 minutes period). 4 / 15
5 Exponential distribution A continous random variable X has exponential distribution with parameter λ > 0 if We denote this by writing X Exp(λ). f X (x) = λe λx for x > 0. Properties: E[X] = λ 1, Var[X] = λ 2. F X (x) = P(X x) = 1 e λx for x > 0. 5 / 15
6 Exponential distribution A continuous random variable X has exponential distribution, if and only if it is memoryless, that is P(X > s + t X > t) = P(X > s), for any s, t > 0. Proof. Suppose X has exponential distribution with parameter λ. Then, we have for any x > 0, and hence P(X > x) = 1 P(X x) = e λx, P(X > s + t) = P(X > s) P(X > t) by the property of the exponential function. The opposite implication follows since the exponential function is the only one satisfying this property, and hence the CDF function is uniquely identified. The only discrete memoryless distribution is the geometric distribution. 6 / 15
7 Exponential distribution Let X be the time until the arrival of the first call at the call center. Since we assume that the arrivals are independent in disjoint intervals, the knowledge that there was no call until time s (this is the event {X > t}, has no influence on the arrivals after time s). Moreover, since the arrivals after time s in an interval of length t should be distributed as the arrivals in the interval [0, t], we get P(X > s + t X > t) = P(X > s), for any s, t > 0, and hence X has exponential distribution. Time until a radioactive particle decays can be modelled with exponential distribution. Time between server requests can be modelled with exponential distribution. 7 / 15
8 Counting process A counting process is stochastic process {C t } t 0 that keeps count of events that have occurred up to time t. In particular, C t it takes values in {0, 1, 2,...}, C t it is non-decreasing in t, C t C s is the number of events in the interval (s, t] for s < t. Here, the word event has the traditional, and not probabilistic, meaning. Counting processes are also called jump processes. 8 / 15
9 Counting process The graph below shows an instance of a counting process together with the times when the events occurred: / 15
10 Poisson process A Poisson process {N t } t 0 with intensity parameter λ > 0 is a counting process with the following additional properties: (starting at 0) N 0 = 0. (independent increments) The random variables (N t2 N t1 ) and (N t4 N t3 ) are independent for all t 1 < t 2 < t 3 < t 4. (stationarity) Fix r > 0. The random variable (N t+r N t ) has the same distribution for all t. (no double jumps/events) P(N h = 1) h λ, and P(N h 2) h 0, as h 0. The last property says that the probability that two events counted by the Poisson process happen at the same time is / 15
11 Poisson process Calls arriving at a call center can be modelled by a Poisson process. Request arriving at a server can be modelled by a Poisson process. Arrivals of students for this lecture cannot be modelled by a Poisson process. 11 / 15
12 Properties of Poisson processes If {N t } t 0 is a Poisson process with intensity λ, then: N t Pois(λt) for all t. In particular, E[N t ] = λt. N t N s Pois(λ(t s)) for all s < t. In particular, E[N t N s ] = λ(t s). Let T 0 = 0, T 1, T 2, T 3,... be the (random) times when the events counted by the process occur. Let τ i = T i T i 1, for i = 1, 2, 3,... Then τ i Exp(λ) for all i, and τ i, τ j are independent for i j. In particular, E[τ i ] = λ 1 for all i. The larger the intensity λ, the more events counted by the process. 12 / 15
13 It is a slow day at a call center of a taxi company. There is 1 call every 10 minutes on average. Bob is the only person present and he needs to use the restroom every hour for 5 minutes. How many calls on average will he miss if he works 8 hours? What is the probability that he will not miss any call? What is the probability that he will miss more than 2 calls? Use a Poisson process to answer the questions. Let X 1, X 2,..., X 8 be the numbers of calls that arrive when Bob is in the restroom each hour. By the properties of the Poisson process, they are all independent, and all have the same distribution Pois(1/2). Hence, by the property of the Poisson distribution X = X 1 + X X 8 Pois(8 1/2) = Pois(4). Therefore, Bob will miss E[X] = 4 calls on average, and P(X = 0) = 40 0! e 4 = e , and P(X > 2) = 1 P(X 2) = = ! e ! e ! e 4 = 1 ( )e / 15
14 Bob is waiting for the first call already 10 minutes. What is the probability that he will still wait more than 20 minutes? Let τ 1 be the time that Bob waits for the first call. By the property of the Poisson process τ 1 has exponential distribution with parameter 1 (here one unit is 10 minutes). We have P(τ 1 > τ 1 > 1) = P(τ 1 > 2) = 1 P(τ 1 2) = e / 15
Lecture 8. Confidence intervals and the central limit theorem
Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of
More informationImportant Probability Distributions OPRE 6301
Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.
More informationExample: 1. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day.
16 The Exponential Distribution Example: 1. You have observed that the number of hits to your web site follow a Poisson distribution at a rate of 2 per day. Let T be the time (in days) between hits. 2.
More informationMAS108 Probability I
1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationLECTURE 16. Readings: Section 5.1. Lecture outline. Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process
LECTURE 16 Readings: Section 5.1 Lecture outline Random processes Definition of the Bernoulli process Basic properties of the Bernoulli process Number of successes Distribution of interarrival times The
More information2. Discrete random variables
2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be
More informationOverview of Monte Carlo Simulation, Probability Review and Introduction to Matlab
Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?
More informationECE302 Spring 2006 HW3 Solutions February 2, 2006 1
ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationIntroduction to Probability
Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence
More information5. Continuous Random Variables
5. Continuous Random Variables Continuous random variables can take any value in an interval. They are used to model physical characteristics such as time, length, position, etc. Examples (i) Let X be
More informationHomework set 4 - Solutions
Homework set 4 - Solutions Math 495 Renato Feres Problems R for continuous time Markov chains The sequence of random variables of a Markov chain may represent the states of a random system recorded at
More informationHow To Find Out How Much Money You Get From A Car Insurance Claim
Chapter 11. Poisson processes. Section 11.4. Superposition and decomposition of a Poisson process. Extract from: Arcones Fall 2009 Edition, available at http://www.actexmadriver.com/ 1/18 Superposition
More informationLecture Notes 1. Brief Review of Basic Probability
Probability Review Lecture Notes Brief Review of Basic Probability I assume you know basic probability. Chapters -3 are a review. I will assume you have read and understood Chapters -3. Here is a very
More informationStochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations
56 Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Florin-Cătălin ENACHE
More informationECE302 Spring 2006 HW4 Solutions February 6, 2006 1
ECE302 Spring 2006 HW4 Solutions February 6, 2006 1 Solutions to HW4 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in
More informationDefinition: Suppose that two random variables, either continuous or discrete, X and Y have joint density
HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,
More informationExponential Distribution
Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1
More informationJoint Exam 1/P Sample Exam 1
Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question
More informationPoisson process. Etienne Pardoux. Aix Marseille Université. Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8
Poisson process Etienne Pardoux Aix Marseille Université Etienne Pardoux (AMU) CIMPA, Ziguinchor 1 / 8 The standard Poisson process Let λ > be given. A rate λ Poisson (counting) process is defined as P
More informationTenth Problem Assignment
EECS 40 Due on April 6, 007 PROBLEM (8 points) Dave is taking a multiple-choice exam. You may assume that the number of questions is infinite. Simultaneously, but independently, his conscious and subconscious
More informatione.g. arrival of a customer to a service station or breakdown of a component in some system.
Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be
More informationMath 431 An Introduction to Probability. Final Exam Solutions
Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <
More informationPoisson Processes. Chapter 5. 5.1 Exponential Distribution. The gamma function is defined by. Γ(α) = t α 1 e t dt, α > 0.
Chapter 5 Poisson Processes 5.1 Exponential Distribution The gamma function is defined by Γ(α) = t α 1 e t dt, α >. Theorem 5.1. The gamma function satisfies the following properties: (a) For each α >
More informationThe Exponential Distribution
21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More information6.041/6.431 Spring 2008 Quiz 2 Wednesday, April 16, 7:30-9:30 PM. SOLUTIONS
6.4/6.43 Spring 28 Quiz 2 Wednesday, April 6, 7:3-9:3 PM. SOLUTIONS Name: Recitation Instructor: TA: 6.4/6.43: Question Part Score Out of 3 all 36 2 a 4 b 5 c 5 d 8 e 5 f 6 3 a 4 b 6 c 6 d 6 e 6 Total
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationIEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS
IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationLecture 5 : The Poisson Distribution
Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,
More informationCHAPTER 6: Continuous Uniform Distribution: 6.1. Definition: The density function of the continuous random variable X on the interval [A, B] is.
Some Continuous Probability Distributions CHAPTER 6: Continuous Uniform Distribution: 6. Definition: The density function of the continuous random variable X on the interval [A, B] is B A A x B f(x; A,
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationRandom variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.
Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()
More informationSums of Independent Random Variables
Chapter 7 Sums of Independent Random Variables 7.1 Sums of Discrete Random Variables In this chapter we turn to the important question of determining the distribution of a sum of independent random variables
More informationMath 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions
Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,
More informationStatistics 100A Homework 4 Solutions
Chapter 4 Statistics 00A Homework 4 Solutions Ryan Rosario 39. A ball is drawn from an urn containing 3 white and 3 black balls. After the ball is drawn, it is then replaced and another ball is drawn.
More informationM/M/1 and M/M/m Queueing Systems
M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential
More information2 Binomial, Poisson, Normal Distribution
2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability
More information6.2. Discrete Probability Distributions
6.2. Discrete Probability Distributions Discrete Uniform distribution (diskreetti tasajakauma) A random variable X follows the dicrete uniform distribution on the interval [a, a+1,..., b], if it may attain
More informationStat 515 Midterm Examination II April 6, 2010 (9:30 a.m. - 10:45 a.m.)
Name: Stat 515 Midterm Examination II April 6, 2010 (9:30 a.m. - 10:45 a.m.) The total score is 100 points. Instructions: There are six questions. Each one is worth 20 points. TA will grade the best five
More informationChapter 4 Lecture Notes
Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,
More informationStatistics 100A Homework 7 Solutions
Chapter 6 Statistics A Homework 7 Solutions Ryan Rosario. A television store owner figures that 45 percent of the customers entering his store will purchase an ordinary television set, 5 percent will purchase
More informationModelling Patient Flow in an Emergency Department
Modelling Patient Flow in an Emergency Department Mark Fackrell Department of Mathematics and Statistics The University of Melbourne Motivation Eastern Health closes more beds Bed closures to cause ambulance
More informationIEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem
IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have
More informationWald s Identity. by Jeffery Hein. Dartmouth College, Math 100
Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,
More informationSection 5.1 Continuous Random Variables: Introduction
Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,
More informationFeb 7 Homework Solutions Math 151, Winter 2012. Chapter 4 Problems (pages 172-179)
Feb 7 Homework Solutions Math 151, Winter 2012 Chapter Problems (pages 172-179) Problem 3 Three dice are rolled. By assuming that each of the 6 3 216 possible outcomes is equally likely, find the probabilities
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationChapter 4. Probability Distributions
Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive
More informationCHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS
CHAPTER 7 SECTION 5: RANDOM VARIABLES AND DISCRETE PROBABILITY DISTRIBUTIONS TRUE/FALSE 235. The Poisson probability distribution is a continuous probability distribution. F 236. In a Poisson distribution,
More informationLecture 7: Continuous Random Variables
Lecture 7: Continuous Random Variables 21 September 2005 1 Our First Continuous Random Variable The back of the lecture hall is roughly 10 meters across. Suppose it were exactly 10 meters, and consider
More informationJan Vecer Tomoyuki Ichiba Mladen Laudanovic. Harvard University, September 29, 2007
Sports Sports Tomoyuki Ichiba Mladen Laudanovic Department of Statistics, Columbia University, http://www.stat.columbia.edu/ vecer Harvard University, September 29, 2007 Abstract Sports In this talk we
More informationANALYZING NETWORK TRAFFIC FOR MALICIOUS ACTIVITY
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 12, Number 4, Winter 2004 ANALYZING NETWORK TRAFFIC FOR MALICIOUS ACTIVITY SURREY KIM, 1 SONG LI, 2 HONGWEI LONG 3 AND RANDALL PYKE Based on work carried out
More informationuation of a Poisson Process for Emergency Care
QUEUEING THEORY WITH APPLICATIONS AND SPECIAL CONSIDERATION TO EMERGENCY CARE JAMES KEESLING. Introduction Much that is essential in modern life would not be possible without queueing theory. All communication
More informationMath 461 Fall 2006 Test 2 Solutions
Math 461 Fall 2006 Test 2 Solutions Total points: 100. Do all questions. Explain all answers. No notes, books, or electronic devices. 1. [105+5 points] Assume X Exponential(λ). Justify the following two
More informationSTAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE
STAT 315: HOW TO CHOOSE A DISTRIBUTION FOR A RANDOM VARIABLE TROY BUTLER 1. Random variables and distributions We are often presented with descriptions of problems involving some level of uncertainty about
More informationPractice Problems #4
Practice Problems #4 PRACTICE PROBLEMS FOR HOMEWORK 4 (1) Read section 2.5 of the text. (2) Solve the practice problems below. (3) Open Homework Assignment #4, solve the problems, and submit multiple-choice
More informationWeek 2: Exponential Functions
Week 2: Exponential Functions Goals: Introduce exponential functions Study the compounded interest and introduce the number e Suggested Textbook Readings: Chapter 4: 4.1, and Chapter 5: 5.1. Practice Problems:
More informationMarshall-Olkin distributions and portfolio credit risk
Marshall-Olkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und
More informationPoisson processes (and mixture distributions)
Poisson processes (and mixture distributions) James W. Daniel Austin Actuarial Seminars www.actuarialseminars.com June 26, 2008 c Copyright 2007 by James W. Daniel; reproduction in whole or in part without
More informationSome special discrete probability distributions
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that
More informationChapter 5 Discrete Probability Distribution. Learning objectives
Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able
More informationChapter 5. Random variables
Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationHomework 4 - KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.
Homework 4 - KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 2-1 Since there can be anywhere from 0 to 4 aces, the
More information6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory
6.263/16.37: Lectures 5 & 6 Introduction to Queueing Theory Massachusetts Institute of Technology Slide 1 Packet Switched Networks Messages broken into Packets that are routed To their destination PS PS
More information), 35% use extra unleaded gas ( A
. At a certain gas station, 4% of the customers use regular unleaded gas ( A ), % use extra unleaded gas ( A ), and % use premium unleaded gas ( A ). Of those customers using regular gas, onl % fill their
More informationProbability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X
Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random
More informationQueueing Systems. Ivo Adan and Jacques Resing
Queueing Systems Ivo Adan and Jacques Resing Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands March 26, 2015 Contents
More informationLECTURE - 1 INTRODUCTION TO QUEUING SYSTEM
LECTURE - 1 INTRODUCTION TO QUEUING SYSTEM Learning objective To introduce features of queuing system 9.1 Queue or Waiting lines Customers waiting to get service from server are represented by queue and
More informationCh5: Discrete Probability Distributions Section 5-1: Probability Distribution
Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.
More informationMilitary Reliability Modeling William P. Fox, Steven B. Horton
Military Reliability Modeling William P. Fox, Steven B. Horton Introduction You are an infantry rifle platoon leader. Your platoon is occupying a battle position and has been ordered to establish an observation
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 18. A Brief Introduction to Continuous Probability
CS 7 Discrete Mathematics and Probability Theory Fall 29 Satish Rao, David Tse Note 8 A Brief Introduction to Continuous Probability Up to now we have focused exclusively on discrete probability spaces
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationThe Normal Distribution. Alan T. Arnholt Department of Mathematical Sciences Appalachian State University
The Normal Distribution Alan T. Arnholt Department of Mathematical Sciences Appalachian State University arnholt@math.appstate.edu Spring 2006 R Notes 1 Copyright c 2006 Alan T. Arnholt 2 Continuous Random
More informationProbability Generating Functions
page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence
More informationHow to Gamble If You Must
How to Gamble If You Must Kyle Siegrist Department of Mathematical Sciences University of Alabama in Huntsville Abstract In red and black, a player bets, at even stakes, on a sequence of independent games
More informationSTAT 3502. x 0 < x < 1
Solution - Assignment # STAT 350 Total mark=100 1. A large industrial firm purchases several new word processors at the end of each year, the exact number depending on the frequency of repairs in the previous
More information10 GEOMETRIC DISTRIBUTION EXAMPLES:
10 GEOMETRIC DISTRIBUTION EXAMPLES: 1. Terminals on an on-line computer system are attached to a communication line to the central computer system. The probability that any terminal is ready to transmit
More information8.7 Exponential Growth and Decay
Section 8.7 Exponential Growth and Decay 847 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound
More informationChapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution
Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover
More informationPROBABILITY AND SAMPLING DISTRIBUTIONS
PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 seema@iasri.res.in. Introduction The concept of probability
More information2WB05 Simulation Lecture 8: Generating random variables
2WB05 Simulation Lecture 8: Generating random variables Marko Boon http://www.win.tue.nl/courses/2wb05 January 7, 2013 Outline 2/36 1. How do we generate random variables? 2. Fitting distributions Generating
More informationRandom Variables. Chapter 2. Random Variables 1
Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets
More informationQuestion: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?
ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the
More informationGambling and Data Compression
Gambling and Data Compression Gambling. Horse Race Definition The wealth relative S(X) = b(x)o(x) is the factor by which the gambler s wealth grows if horse X wins the race, where b(x) is the fraction
More information6 POISSON DISTRIBUTIONS
6 POISSON DISTRIBUTIONS Chapter 6 Poisson Distributions Objectives After studying this chapter you should be able to recognise when to use the Poisson distribution; be able to apply the Poisson distribution
More informationSolutions to Problem Set 1
Cornell University, Physics Department Fall 2013 PHYS-3341 Statistical Physics Prof. Itai Cohen Solutions to Problem Set 1 David C. Tsang, Woosong Choi, Philip Kidd, Igor Segota, Yariv Yanay 1.3 Birthday
More informationUNIT 2 QUEUING THEORY
UNIT 2 QUEUING THEORY LESSON 24 Learning Objective: Apply formulae to find solution that will predict the behaviour of the single server model II. Apply formulae to find solution that will predict the
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 13. Random Variables: Distribution and Expectation
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 3 Random Variables: Distribution and Expectation Random Variables Question: The homeworks of 20 students are collected
More informationChapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions
Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the
More informationECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003
ECON1003: Analysis of Economic Data Fall 2003 Answers to Quiz #2 11:40a.m. 12:25p.m. (45 minutes) Tuesday, October 28, 2003 1. (4 points) The number of claims for missing baggage for a well-known airline
More informationChapter 9 Monté Carlo Simulation
MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?
More informationTEST 2 STUDY GUIDE. 1. Consider the data shown below.
2006 by The Arizona Board of Regents for The University of Arizona All rights reserved Business Mathematics I TEST 2 STUDY GUIDE 1 Consider the data shown below (a) Fill in the Frequency and Relative Frequency
More informationCHAPTER 3 CALL CENTER QUEUING MODEL WITH LOGNORMAL SERVICE TIME DISTRIBUTION
31 CHAPTER 3 CALL CENTER QUEUING MODEL WITH LOGNORMAL SERVICE TIME DISTRIBUTION 3.1 INTRODUCTION In this chapter, construction of queuing model with non-exponential service time distribution, performance
More informationNovember 2000 Course 3
November Course 3 Society of Actuaries/Casualty Actuarial Society November - - GO ON TO NEXT PAGE Questions through 36 are each worth points; questions 37 through 44 are each worth point.. For independent
More information. (3.3) n Note that supremum (3.2) must occur at one of the observed values x i or to the left of x i.
Chapter 3 Kolmogorov-Smirnov Tests There are many situations where experimenters need to know what is the distribution of the population of their interest. For example, if they want to use a parametric
More informationBasic Queueing Theory
Basic Queueing Theory Dr. János Sztrik University of Debrecen, Faculty of Informatics Reviewers: Dr. József Bíró Doctor of the Hungarian Academy of Sciences, Full Professor Budapest University of Technology
More informationThe Binomial Probability Distribution
The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability
More informationCharacteristics of Binomial Distributions
Lesson2 Characteristics of Binomial Distributions In the last lesson, you constructed several binomial distributions, observed their shapes, and estimated their means and standard deviations. In Investigation
More information