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.


 Daniela Hilda West
 1 years ago
 Views:
Transcription
1 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. You observe the number of calls that arrive each day over a period of a year, and note that the arrivals follow a Poisson distribution with an average of 3 per day. Let T be the waiting time between calls. 3. Records show that job submissions to a busy computer centre have a Poisson distribution with an average of 4 per minute. Let T be the time in minutes between submissions. 4. Records indicate that messages arrive to a computer server following a Poisson distribution at the rate of 6 per hour. Let T be the time in hours that elapses between messages.
2 Probability Density Function Of Waiting Times Generally the exponential distribution describes waiting time between Poisson occurrences Proof: Let T = time that elapses after a Poisson event. P(T > t) = probability that no event occurred in the time interval of length t. The probability that no Poisson event occurred in the time interval [, t]: P(, t) = e λt. where λ is the average Poisson occurrence rate in a unit time interval. So: P(T > t) = e λt, Hence the CDF is: F(t) = P(T t) = 1 e λt and, working backwards, the PDF is f(t) = F (t) = λe λt
3 The PDF: f(t) = λe λt, t > = otherwise The CDF: P(T t) = F(t) = t λe λt dt = [ e λt ] t T= = e λt + 1 i.e. F(t) = 1 e λt, t > = otherwise.
4 Example: If jobs arrive every 15 seconds on average, λ = 4 per minute, what is the probability of waiting less than or equal to 3 seconds, i.e.5 min? P(T.5). dexp(x, 4) P (X <=.5) x P(T.5) =.5 4e 4t dt = [ e 4t].5 t= = 1 e 2 =.86 From R pexp(.5,4) [1]
5 What is the maximum waiting time between two job submissions with 95% confidence? We need to find k so that This is the quantile function. qexp(.95, 4) [1] P(T k) =.95 The probability that there will be.74 min, about 45 seconds, between two job submissions is.95. Applications of the Exponential Distribution: 1. Time between telephone calls 2. Time between machine breakdowns 3. Time between successive job arrivals at a computing centre
6 Example Accidents occur with a Poisson distribution at an average of 4 per week. i.e. λ = 4 1. Calculate the probability of more than 5 accidents in any one week 2. What is the probability that at least two weeks will elapse between accident? Solution 1. Poisson: In R 1ppois(5, 4) [1] Exponential: P(X > 5) = 1 P(X 5) P(Time between occurrences > 2) = 2 λe λt dt In R = [ e λt ] T=2 = e 8 =.34 pexp(2, 4) [1] pexp(2, 4) [1]
7 Density of the Exponential Distribution with λ = 2, 3, 4 and 6 lambda = 2 lambda = 3 dexp(x, 2) dexp(x, 3) x x lambda = 4 lambda = 6 dexp(x, 4) 2 4 dexp(x, 6) x x par(mfrow = c(2,2)) curve(dexp(x, 2),, 3, main ="lambda = 2") curve(dexp(x, 3),, 3, main ="lambda = 3") curve(dexp(x, 4),, 3, main ="lambda = 4") curve(dexp(x, 6),, 3, main ="lambda = 6")
8 The Markov Property of Exponential Examples: 1. The distribution of the remaining life does not depend on how long the component has been operating. i.e. the component does not age  its breakdown is a result of some sudden failure not a gradual deterioration 2. Time between telephone calls Waiting time for a call is independent of how long we have been waiting
9 The Markov property Show: P(T x + t T > t) = P(T x) Proof: E 1 = T x + t, and E 2 = T > t Then: Now P(E 1 E 2 ) = P(E 1 E 2 ) P(E 2 ) P(E 1 E 2 ) = P(t < T x + t) = x+t t λe λt dt = e λt [1 e λx ] and thus now P(E 2 ) = t λe λt dt = e λt P(E 1 E 2 ) = e λt [1 e λx ] e λt = 1 e λx 1 e λx = F(x) = P(T x)
10 Examples 1. Calls arrive at an average rate of 12 per hour. Find the probability that a call will occur in the next 5 minutes given that you have already waited 1 minutes for a call i.e. Find P(T 15 T > 1) From the Markov property P(T 15 T > 1) = P(T 5) So: P(T 5) = 1 e 5λ The average rate of telephone calls is λ = 2 in a minute, then P(T 5) = 1 e (5)( 2) = 1 e 1 = 1.37 =.63 In R > pexp(5,.2) [1]
11 Examples 2. The average rate of job submissions in a busy computer centre is 4 per minute. If it can be assumed that the number of submissions per minute interval is Poisson distributed, calculate the probability that: (a) at least 15 seconds will elapse between any two jobs. (b) less than 1 minutes will elapse between jobs. (c) If no jobs have arrived in the last 3 seconds, what is the probability that a job will arrive in the next 15 seconds? Solution λ = 4 per minute (a) P(t > 15 sec.) = P(T >.25 min) =.25 λe λt dt = [ e λt ] T=.25 = e 1 =.37 In R > 1 pexp(.25, 4) [1]
12 Mean of the Exponential Distribution Recall that when X is continuous: E(X) = x xf(x)dx For the exponential distribution: E(T) = tλe λt dt Integration by parts: udv = (uv) tλe λt dt = udv Trick is to spot the u and v: vdu Take and u = t dv = λe λt dt which gives v = e λt Then tλe λt dt = ( te λt) + e λt dt = ( te λt λ 1 e λt) = 1 λ
13 R Functions for the Exponential Distribution Density Function: dexp dexp(1, 4) Cumulative Distribution Function pexp pexp(5, 4) P(T 5) with λ = 4 Quantile Function qexp qexp(.95, 4) Choose k so that > qexp(.95, 4) 1] P(T k).95.
Poisson 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 informationMEP Y9 Practice Book A
1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,
More informationSolution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.34.4) Homework Solutions. Section 4.
Math 115 N. Psomas Chapter 4 (Sections 4.34.4) Homework s Section 4.3 4.53 Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give
More informationIntroduction to Queueing Theory and Stochastic Teletraffic Models
Introduction to Queueing Theory and Stochastic Teletraffic Models Moshe Zukerman EE Department, City University of Hong Kong Copyright M. Zukerman c 2000 2015 Preface The aim of this textbook is to provide
More informationFindTheNumber. 1 FindTheNumber With Comps
FindTheNumber 1 FindTheNumber With Comps Consider the following twoperson game, which we call FindTheNumber with Comps. Player A (for answerer) has a number x between 1 and 1000. Player Q (for questioner)
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationFinding the Payment $20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = $488.26
Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive $5,000 per month in retirement.
More informationPractical Driving Assessment bookings online Frequently Asked Questions
Practical Driving Assessment bookings online Frequently Asked Questions How can I book a Practical Driving Assessment (PDA)? Bookings can be made: In person at a Driver and Vehicle Services Centre (between
More informationIntroduction to Differential Calculus. Christopher Thomas
Mathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationPAYE (Pay as you Earn) Income Tax
Income Tax and National Insurance There are two types of tax on the income you receive as an employee in the United Kingdom Income Tax and National Insurance. They are calculated using different methods
More informationONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK
ONEDIMENSIONAL RANDOM WALKS 1. SIMPLE RANDOM WALK Definition 1. A random walk on the integers with step distribution F and initial state x is a sequence S n of random variables whose increments are independent,
More information( ) where W is work, f(x) is force as a function of distance, and x is distance.
Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work
More informationfor Managers and Admins
Need to change the steps in a business process to match the way your organization does things? Read this guide! for Managers and Admins contents What is a business process? The basics of customizing a
More informationUSING A TI83 OR TI84 SERIES GRAPHING CALCULATOR IN AN INTRODUCTORY STATISTICS CLASS
USING A TI83 OR TI84 SERIES GRAPHING CALCULATOR IN AN INTRODUCTORY STATISTICS CLASS W. SCOTT STREET, IV DEPARTMENT OF STATISTICAL SCIENCES & OPERATIONS RESEARCH VIRGINIA COMMONWEALTH UNIVERSITY Table
More informationSTATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS
STATISTICS 8: CHAPTERS 7 TO 10, SAMPLE MULTIPLE CHOICE QUESTIONS 1. If two events (both with probability greater than 0) are mutually exclusive, then: A. They also must be independent. B. They also could
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationProgramming Your Calculator Casio fx7400g PLUS
Programming Your Calculator Casio fx7400g PLUS Barry Kissane Programming Your Calculator: Casio fx7400g PLUS Published by Shriro Australia Pty Limited 7274 Gibbes Street, Chatswood NSW 2067, Australia
More informationGuide for Texas Instruments TI83, TI83 Plus, or TI84 Plus Graphing Calculator
Guide for Texas Instruments TI83, TI83 Plus, or TI84 Plus Graphing Calculator This Guide is designed to offer stepbystep instruction for using your TI83, TI83 Plus, or TI84 Plus graphing calculator
More informationInsurance in your super
Insurance in your super The information in this document forms part of the: UniSuper Accumulation 1 Product Disclosure Statement issued on 3 January 2015 UniSuper Defined Benefit Division and Accumulation
More informationSelecting a Subset of Cases in SPSS: The Select Cases Command
Selecting a Subset of Cases in SPSS: The Select Cases Command When analyzing a data file in SPSS, all cases with valid values for the relevant variable(s) are used. If I opened the 1991 U.S. General Social
More informationIntroduction. Continue to Step 1: Creating a Process Simulator Model. Goal: 40 units/week of new component. Process Simulator Tutorial
Introduction This tutorial places you in the position of a process manager for a specialty electronics manufacturing firm that makes small lots of prototype boards for medical device manufacturers. Your
More informationAbout the Return on Investment of TestDriven Development
About the Return on of TestDriven Development Matthias M. Müller Fakultät für Informatik Universität Karlsruhe, Germany muellerm @ ira.uka.de Frank Padberg Fakultät für Informatik Universität Karlsruhe,
More informationUnit 26: Small Sample Inference for One Mean
Unit 26: Small Sample Inference for One Mean Prerequisites Students need the background on confidence intervals and significance tests covered in Units 24 and 25. Additional Topic Coverage Additional coverage
More informationSpatial Point Processes and their Applications
Spatial Point Processes and their Applications Adrian Baddeley School of Mathematics & Statistics, University of Western Australia Nedlands WA 6009, Australia email: adrian@maths.uwa.edu.au A spatial
More informationBPMN by example. Bizagi Suite. Copyright 2014 Bizagi
BPMN by example Bizagi Suite Recruitment and Selection 1 Table of Contents Scope... 2 BPMN 2.0 Business Process Modeling Notation... 2 Why Is It Important To Model With Bpmn?... 2 Introduction to BPMN...
More informationInteraction effects between continuous variables (Optional)
Interaction effects between continuous variables (Optional) Richard Williams, University of Notre Dame, http://www.nd.edu/~rwilliam/ Last revised February 0, 05 This is a very brief overview of this somewhat
More information