Bonusmalus systems and Markov chains


 Osborn Lee
 2 years ago
 Views:
Transcription
1 Bonusmalus systems and Markov chains Dutch car insurance bonusmalus system class % increase new class after # claims > New insurants enter at stage 100%. The second column represents the percentage increase / decrease of the basic premium (based on rating factors like price of the car, horsepower of the car, etc.). Definition and basic properties of Markov chains Definition: A stochastic process X = {X t, t } in discrete time is called a (first order) Markov chain if P (X t+1 = j X t = i, X t 1 = i t 1,..., X 0 = i 0 ) = P (X t+1 = j X t = i) for all j, i, i t 1,..., i 0 E, where E is the state space of the Markov chain. The probability p ij (t) = P (X t+1 = j X t = i) is called (one step) transition probability. The matrix P (t) = (p ij (t)) 1
2 is called transition matrix or matrix of transition probabilities. It is a stochastic matrix, since all rows sum to one. If p ij (t) p ij (and therefore P (t) P ), the Markov chain is called homogeneous. The probabilities p (h) ij (t) = P (X t+h = j X t = i) are called hstep transition probabilities, in particular, p ij (t) p ij (t, 1). Similarly, the matrix is called hstep transition matrix. P (h) (t) = (p (h) ij (t)) The distribution of X t+1 is conditionally independent from the past, given the current value X t. The above definition is only valid for discrete valued random variables, i.e. if the state space E is countable. We will only consider discrete Markov chains. Markov chains can be generalised to continuous time and are then called (discrete) Markov processes. Distribution of a Markov chain: Given the transition probabilities p ij and a starting distribution p (0) i of a Markov chain is uniquely determined. = P (X 0 = i), the distribution The hstep transition probabilities are determined by the ChapmanKolmogorov equations: and therefore P (h) = P h = P... P. P (h+l) = P (h) P (l) For the transition probabilities the ChapmanKolmogorov equations can be expressed as p (h+l) ij = k E p (h) ik p(l) kj. Based on the hstep transition probabilities, we obtain P (X tn = i n, X tn 1 = i n 1,..., X t1 = i 1, X 0 = i 0 ) = p (0) i 0 p (t 1) i 0 i 1 p (t 2 t 1 ) i 1 i 2... p (tn t n 1) i n 1 i n Example: BonusMalus system BonusMalus systems can be considered as homogeneous Markov chains with a finite state space E of bonus malus classes. Suppose that the l classes are ordered such that the corresponding premiums are decreasing, i.e. π 1 π 2... π l. The first class is sometimes called super malus class and the last class super bonus class. 2
3 Frequently a Poisson distribution is used to model the transition probabilities within a bonusmalus system. To be more specific, the Poisson distribution describes the number of claims for an individual and the transition probabilities are determined from this claim frequency distribution. class German bonusmalus system (up to 2002) premium new class after # claims rate >
4 4 where 1.. = 0 Transition matrix for the German bonusmalus system (up to 2002) i/j {0}..... {1}... {2}. {3}.... {4, 5,...} 2 {0}..... {1}... {2}. {3}.... {4, 5,...} 3. {0}..... {1}.. {2}. {3}.... {4, 5,...} 4.. {0}..... {1}. {2}. {3}.... {4, 5,...} 5... {0}..... {1}. {2} {3}.... {4, 5,...} {0}..... {1}. {2} {3}... {4, 5,...} {0}.... {1}. {2} {3}... {4, 5,...} {0}... {1}. {2} {3}... {4, 5,...} {0}.. {1}. {2} {3}... {4, 5,...} {0}.. {1} {2} {3}... {4, 5,...} {0}.. {1} {2}. {3}. {4, 5,...} {0}. {1} {2}. {3}. {4, 5,...} {0}. {1}. {2} {3} {4, 5,...} {0}.. {1} {2} {3, 4,...} {0}.. {1} {2} {3, 4,...} {0}... {1} {2, 3,...} {0}.... {1, 2,...} {0}.... {1, 2,...} 2. {k} = p k = λk k! e λ. 3. {k, k + 1,...} = i=k p i Supplementary material (Risk Theory) Summer term 2007
5 1. The classes with premium rate 100% are called bonus classes, the classes with premium rate > 100% are malus classes. 2. In the German bonusmalus system, the classes are called Schadenfreiheitsklassen. 3. Since 2003 Germany has a new bonusmalus system with 23 bonus classes and 3 malus classes (and reversed order as in the Dutch example from above). Stationary distribution of a Markov chain Under regularity conditions on the transition matrix P, the linear systems of equations µ = µp has a unique, strictly nonzero solution µ, where µ = (µ j, j E). This solution µ is called the stationary distribution of the Markov chain, since µ j = lim p (t) ij, t and for any starting distribution. lim t p(0) P t = p (0) P = µ 1. If p (0) = µ, then P p (0) = µ, i.e. the state probabilities are timeconstant. This is the reason why µ is called stationary distribution. 2. The stationary distribution can be obtained as µ = (I P + Q) 1 where = (1,..., 1), I is the identity matrix and Q = In a bonusmalus system, the stationary distribution represents the percentage of drivers in the different classes after the bonusmalus system has been run for a long time. 5
B. Maddah ENMG 501 Engineering Management I 03/17/07
B. Maddah ENMG 501 Engineering Management I 03/17/07 Discrete Time Markov Chains (3) Longrun Properties of MC (Stationary Solution) Consider the twostate MC of the weather condition in Example 4. P 0.5749
More informationTHE BONUSMALUS SYSTEM MODELLING USING THE TRANSITION MATRIX
1502 Challenges of the Knowledge Society Economics THE BONUSMALUS SYSTEM MODELLING USING THE TRANSITION MATRIX SANDRA TEODORESCU * Abstract The motor insurance is an important branch of nonlife insurance
More informationLECTURE 4. Last time: Lecture outline
LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random
More informationIntroduction to Probability Theory for Graduate Economics. Fall 2008
Introduction to Probability Theory for Graduate Economics Fall 2008 Yiğit Sağlam December 1, 2008 CHAPTER 5  STOCHASTIC PROCESSES 1 Stochastic Processes A stochastic process, or sometimes a random process,
More informationMarkov Chains, Stochastic Processes, and Advanced Matrix Decomposition
Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms
More informationConstructing Phylogenetic Trees Using Maximum Likelihood
Claremont Colleges Scholarship @ Claremont Scripps Senior Theses Scripps Student Scholarship 2012 Constructing Phylogenetic Trees Using Maximum Likelihood Anna Cho Scripps College Recommended Citation
More informationReduced echelon form: Add the following conditions to conditions 1, 2, and 3 above:
Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row is in
More informationMarkov Chain Monte Carlo Simulation Made Simple
Markov Chain Monte Carlo Simulation Made Simple Alastair Smith Department of Politics New York University April2,2003 1 Markov Chain Monte Carlo (MCMC) simualtion is a powerful technique to perform numerical
More informationAn Extension Model of FinanciallyBalanced BonusMalus System
An Extension Model of FinanciallyBalanced BonusMalus System Xiao Yugu 1), Meng Shengwang 1), Robert Conger 2) 1. School of Statistics, Renmin University of China 2. Towers Perrin, Chicago Contents Introduction
More information1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easytocheck
More information15 Markov Chains: Limiting Probabilities
MARKOV CHAINS: LIMITING PROBABILITIES 67 Markov Chains: Limiting Probabilities Example Assume that the transition matrix is given by 7 2 P = 6 Recall that the nstep transition probabilities are given
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationIntroduction to Stationary Distributions
13 Introduction to Stationary Distributions We first briefly review the classification of states in a Markov chain with a quick example and then begin the discussion of the important notion of stationary
More information12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
More informationBonusMalus System in Iran: An Empirical Evaluation
Journal of Data Science 11(2013), 2941 BonusMalus System in Iran: An Empirical Evaluation Rahim Mahmoudvand 1, Alireza Edalati 2 and Farhad Shokoohi 1 1 Shahid Beheshti University and 2 Institut für
More informationAnnouncements. CompSci 230 Discrete Math for Computer Science. Test 1
CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 47 Prof. Rodger is out Sep 30Oct 4 There is Recitation: Sept 2730.
More informationPerformance Analysis of a Telephone System with both Patient and Impatient Customers
Performance Analysis of a Telephone System with both Patient and Impatient Customers Yiqiang Quennel Zhao Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationIntroduction to Flocking {Stochastic Matrices}
Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur  Yvette May 21, 2012 CRAIG REYNOLDS  1987 BOIDS The Lion King CRAIG REYNOLDS
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationGLM III: Advanced Modeling Strategy 2005 CAS Seminar on Predictive Modeling Duncan Anderson MA FIA Watson Wyatt Worldwide
GLM III: Advanced Modeling Strategy 25 CAS Seminar on Predictive Modeling Duncan Anderson MA FIA Watson Wyatt Worldwide W W W. W A T S O N W Y A T T. C O M Agenda Introduction Testing the link function
More informationCHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction
CHAPTER 7 STOCHASTIC ANALYSIS OF MANPOWER LEVELS AFFECTING BUSINESS 7.1 Introduction Consider in this chapter a business organization under fluctuating conditions of availability of manpower and business
More information6. Jointly Distributed Random Variables
6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.
More informationSYSTEMS OF EQUATIONS
SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which
More informationDPolicy for a Production Inventory System with Perishable Items
Chapter 6 DPolicy for a Production Inventory System with Perishable Items 6.1 Introduction So far we were concentrating on invenory with positive (random) service time. In this chapter we concentrate
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More information4. Joint Distributions
Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose
More information3.2 Roulette and Markov Chains
238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationA linear algebraic method for pricing temporary life annuities
A linear algebraic method for pricing temporary life annuities P. Date (joint work with R. Mamon, L. Jalen and I.C. Wang) Department of Mathematical Sciences, Brunel University, London Outline Introduction
More informationSection 6.1 Joint Distribution Functions
Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function
More informationOptimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani. EPEW 2014, Florence
Optimal Hiring of Cloud Servers A. Stephen McGough, Isi Mitrani EPEW 2014, Florence Scenario How many cloud instances should be hired? Requests Host hiring servers The number of active servers is controlled
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationName: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
More informationNotes for STA 437/1005 Methods for Multivariate Data
Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationMath Homework 7 Solutions
Math 450  Homework 7 Solutions 1 Find the invariant distribution of the transition matrix: 0 1 0 P = 2 1 0 3 3 p 1 p 0 The equation πp = π, and the normalization condition π 1 + π 2 + π 3 = 1 give the
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationMatrices, Determinants and Linear Systems
September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we
More informationExam Introduction Mathematical Finance and Insurance
Exam Introduction Mathematical Finance and Insurance Date: January 8, 2013. Duration: 3 hours. This is a closedbook exam. The exam does not use scrap cards. Simple calculators are allowed. The questions
More informationWorked examples Random Processes
Worked examples Random Processes Example 1 Consider patients coming to a doctor s office at random points in time. Let X n denote the time (in hrs) that the n th patient has to wait before being admitted
More informationSome ergodic theorems of linear systems of interacting diffusions
Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ liuyong@math.pku.edu.cn yangfx@math.pku.edu.cn 1 1 30 1 The ergodic theory of interacting systems has
More informationNotes on Probability Theory
Notes on Probability Theory Christopher King Department of Mathematics Northeastern University July 31, 2009 Abstract These notes are intended to give a solid introduction to Probability Theory with a
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
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 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 informationWhat did you expect? Lessons from the French environmental Bonus/Malus
What did you expect? Lessons from the French environmental Bonus/Malus X. d Haultfœuille, I. Durrmeyer, P. Février Environmental bonus/malus policy Main objectives of the policy Reduce CO 2 emissions related
More informationA Markovian Model for Investment Analysis in Advertising
Proceedings of the Seventh Young Statisticians Meeting Andrej Mrvar (Editor) Metodološki zvezki, 21, Ljubljana: FDV, 2003 A Markovian Model for Investment Analysis in Advertising Eugenio Novelli 1 Abstract
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationA DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION
A DISCRETE TIME MARKOV CHAIN MODEL IN SUPERMARKETS FOR A PERIODIC INVENTORY SYSTEM WITH ONE WAY SUBSTITUTION A Class Project for MATH 768: Applied Stochastic Processes Fall 2014 By Chudamani Poudyal &
More information10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method
578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after
More information1 Gaussian Elimination
Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 GaussJordan reduction and the Reduced
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationFrom the probabilities that the company uses to move drivers from state to state the next year, we get the following transition matrix:
MAT 121 Solutions to TakeHome Exam 2 Problem 1 Car Insurance a) The 3 states in this Markov Chain correspond to the 3 groups used by the insurance company to classify their drivers: G 0, G 1, and G 2
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationThese axioms must hold for all vectors ū, v, and w in V and all scalars c and d.
DEFINITION: A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the following axioms
More informationDirect Methods for Solving Linear Systems. Linear Systems of Equations
Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationMarkov Chains and Queueing Networks
CS 797 Independent Study Report on Markov Chains and Queueing Networks By: Vibhu Saujanya Sharma (Roll No. Y211165, CSE, IIT Kanpur) Under the supervision of: Prof. S. K. Iyer (Dept. Of Mathematics, IIT
More informationAviation Infrastructure Economics
Aviation Short Course Aviation Infrastructure Economics October 1415, 15, 2004 The Aerospace Center Building 901 D St. SW, Suite 850 Washington, DC 20024 Lecture BWI/Andrews Conference Rooms Instructor:
More information4.1 Introduction and underlying setup
Chapter 4 SemiMarkov processes in labor market theory 4.1 Introduction and underlying setup SemiMarkov processes are, like all stochastic processes, models of systems or behavior. As extensions of Markov
More informationLecture 6: The Group Inverse
Lecture 6: The Group Inverse The matrix index Let A C n n, k positive integer. Then R(A k+1 ) R(A k ). The index of A, denoted IndA, is the smallest integer k such that R(A k ) = R(A k+1 ), or equivalently,
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationSeminar assignment 1. 1 Part
Seminar assignment 1 Each seminar assignment consists of two parts, one part consisting of 3 5 elementary problems for a maximum of 10 points from each assignment. For the second part consisting of problems
More informationApplication of Markov chain analysis to trend prediction of stock indices Milan Svoboda 1, Ladislav Lukáš 2
Proceedings of 3th International Conference Mathematical Methods in Economics 1 Introduction Application of Markov chain analysis to trend prediction of stock indices Milan Svoboda 1, Ladislav Lukáš 2
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationSTABILITY OF LUKUMAR NETWORKS UNDER LONGESTQUEUE AND LONGESTDOMINATINGQUEUE SCHEDULING
Applied Probability Trust (28 December 2012) STABILITY OF LUKUMAR NETWORKS UNDER LONGESTQUEUE AND LONGESTDOMINATINGQUEUE SCHEDULING RAMTIN PEDARSANI and JEAN WALRAND, University of California, Berkeley
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.2 Row Reduction and Echelon Forms ECHELON FORM A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties: 1. All nonzero
More informationIntroduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem
More informationSimple Markovian Queueing Systems
Chapter 4 Simple Markovian Queueing Systems Poisson arrivals and exponential service make queueing models Markovian that are easy to analyze and get usable results. Historically, these are also the models
More informationMATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.
MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all ndimensional column
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationSTAT/MTHE 353: Probability II. STAT/MTHE 353: Multiple Random Variables. Review. Administrative details. Instructor: TamasLinder
STAT/MTHE 353: Probability II STAT/MTHE 353: Multiple Random Variables Administrative details Instructor: TamasLinder Email: linder@mast.queensu.ca T. Linder ueen s University Winter 2012 O ce: Je ery
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 informationSPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIXEXPONENTIAL DISTRIBUTIONS
Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 15326349 print/15324214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS Linear systems Ax = b occur widely in applied mathematics They occur as direct formulations of real world problems; but more often, they occur as a part of the numerical analysis
More informationWebbased Supplementary Materials for Bayesian Effect Estimation. Accounting for Adjustment Uncertainty by Chi Wang, Giovanni
1 Webbased Supplementary Materials for Bayesian Effect Estimation Accounting for Adjustment Uncertainty by Chi Wang, Giovanni Parmigiani, and Francesca Dominici In Web Appendix A, we provide detailed
More informationMarkov Chains. Chapter 4. 4.1 Stochastic Processes
Chapter 4 Markov Chains 4 Stochastic Processes Often, we need to estimate probabilities using a collection of random variables For example, an actuary may be interested in estimating the probability that
More informationFitting the Belgian BonusMalus System
Fitting the Belgian BonusMalus System S. Pitrebois 1, M. Denuit 2 and J.F. Walhin 3 Abstract. We show in this paper how to obtain the relativities of the Belgian BonusMalus System, including the special
More informationDETERMINANTS. b 2. x 2
DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in
More informationGeneral theory of stochastic processes
CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result
More informationRandom access protocols for channel access. Markov chains and their stability. Laurent Massoulié.
Random access protocols for channel access Markov chains and their stability laurent.massoulie@inria.fr Aloha: the first random access protocol for channel access [Abramson, Hawaii 70] Goal: allow machines
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationSudoku an alternative history
Sudoku an alternative history Peter J. Cameron p.j.cameron@qmul.ac.uk Talk to the Archimedeans, February 2007 Sudoku There s no mathematics involved. Use logic and reasoning to solve the puzzle. Instructions
More informationBindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12
Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer
More informationChapter 6. Linear Programming: The Simplex Method. Introduction to the Big M Method. Section 4 Maximization and Minimization with Problem Constraints
Chapter 6 Linear Programming: The Simplex Method Introduction to the Big M Method In this section, we will present a generalized version of the simplex method that t will solve both maximization i and
More informationMath 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns
Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in
More informationMath 2020 Quizzes Winter 2009
Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile
More informationSystems of Linear Equations
A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Systems of Linear Equations Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION AttributionNonCommercialShareAlike (CC
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely sidestepped
More information1 Sufficient statistics
1 Sufficient statistics A statistic is a function T = rx 1, X 2,, X n of the random sample X 1, X 2,, X n. Examples are X n = 1 n s 2 = = X i, 1 n 1 the sample mean X i X n 2, the sample variance T 1 =
More informationMatrix Calculations: Kernels & Images, Matrix Multiplication
Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers) Institute for Computing and Information Sciences Intelligent Systems Version: spring 2016 A. Kissinger Version:
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More information