Continuous Time Markov Chains
|
|
- Cornelius Webster
- 6 years ago
- Views:
Transcription
1 Contnuous Tme Markov Chans Brth and Death Processes,Transton Probablty Functon, Kolmogorov Equatons, Lmtng Probabltes, Unformzaton Chapter 6 1
2 Markovan Processes State Space Parameter Space (Tme) Dscrete Contnuous Dscrete Markov chans (Chapter 4) Contnuous tme Markov chans (Chapters 5, 6) Contnuous Brownan moton process (Chapter 10) Chapter 6 2
3 Contnuous Tme Markov Chan A stochastc process {X(t), t 0} s a contnuous tme Markov chan (CTMC) f for all s, t 0 and nonnegatve ntegers, j, x(u), 0 u < s, ( ) ( ) ( ) ( ) { + = =, =,0 < } P X s t j X s X u x u u s ( ) ( ) { } = P X s+ t = j X s = and f ths probablty s ndependent of s, then the CTMC has statonary transton probabltes: P t = P X s+ t = j X s = s j () ( ) ( ) { } for all Chapter 6 3
4 Alternate Defnton Each tme the process enters state, The amount of tme t spends n state before makng a transton to a dfferent state s exponentally dstrbuted wth parameter v, and When t leaves state, t next enters state j wth probablty P j, where P = 0 and P = j j 1 Let q = vp, then v = q, j j j j ( h) Pj ( h) 1 P lm = v and lm = q h 0 h h 0 h j Chapter 6 4
5 Brth and Death Processes If a CTMC has states {0, 1, } and transtons from state n may go only to ether state n - 1 or state n + 1, t s called a brth and death process. The brth (death) rate n state n s λ n (µ n ), so v0 = λ0 v = λ + µ, > 0 P01 = 1 λ µ P, + 1 =, P, 1 =, > 0 λ + µ λ + µ λ o λ n-1 λ n λ n-1 n n+1 µ 1 µ 2 µ n µ n+1 Chapter 6 5
6 Chapman-Kolmogorov Equatons In order to get from state at tme 0 to state j at tme t + s, the process must be n some state k at tme t From these can be derved two sets of dfferental equatons: Backward P j ( t) = qk Pkj ( t) vp j ( t) Forward ( ) ( ) ( ) P t s P t P s j k kj k = 0 + = k ( ) ( ) ( ) P t = q P t v P t j kj k j j k j Chapter 6 6
7 Lmtng Probabltes If All states of the CTMC communcate: For each par, j, startng n state there s a postve probablty of ever beng n state j, and The chan s postve recurrent: startng n any state, the expected tme to return to that state s fnte, then lmtng probabltes exst: P = lm P t (and when the lmtng probabltes exst, the chan s called ergodc) Can we fnd them by solvng somethng lke π = π P for dscrete tme Markov chans? j t j ( ) Chapter 6 7
8 Infntesmal Generator (Rate) Matrx Let R be a matrx wth elements (the rows of R sum to 0) Let t qj,f j = v,f = j n the forward equatons. In steady state: lm P t = lm q P t v P t ( ) ( ) ( ) j kj k j j t t k j 0 = q P v P k j These can be wrtten n matrx form as PR = 0 along wth P = j j 1 and solved for the lmtng probabltes. What do you get f you do the same wth the backward equatons? r kj k j j j Chapter 6 8
9 Balance Equatons The PR = 0 equatons can also be nterpreted as balancng: vp = q P j j kj k k j rate at whch process leaves j = rate at whch process enters j For a brth-death process, they are equvalent to levelcrossng equatons λ npn = µ n + 1Pn + 1 rate of crossng from n to n+ 1 = rate of crossng from n+ 1 to n so P n = λ λ λ µµ µ P and a steady state exsts f λλ λn µµ µ 0 1 n n n= n < Chapter 6 9
10 Tme Reversblty A CTMC s tme-reversble f and only f Pq j = Pq j j when j There are two mportant results: 1. An ergodc brth and death process s tme reversble 2. If for some set of numbers {P }, P = 1 and Pq = Pq when j j j j then the CTMC s tme-reversble and P s the lmtng probablty of beng n state. Ths can be a way of fndng the lmtng probabltes. Chapter 6 10
11 Unformzaton Before, we assumed that P = 0,.e., when the process leaves state t always goes to a dfferent state. Now, let v be any number such that v v for all. Assume that all transtons occur at rate v, but that n state, only the fracton v /v of them are real ones that lead to a dfferent state. The rest are fcttous transtons where the process stays n state. Usng ths fcttous rate, the tme the process spends n state s exponental wth rate v. When a transton occurs, t goes to state j wth probablty v 1, j = * v Pj = v P j, j v Chapter 6 11
12 Unformzaton (2) In the unformzed process, the number of transtons up to tme t s a Posson process N(t) wth rate v. Then we can compute the transton probabltes by condtonng on N(t): j { } () () ( 0) P t = P X t = j X = n= 0 n= 0 n= 0 { () ( 0 ), () } () ( 0) { () ( 0 ), () } * n j vt ( vt) n! n { } = P X t = j X = N t = n P N t = n X = = P X t = j X = N t = n = P e e vt ( vt) n! n Chapter 6 12
13 More on the Rate Matrx ( ) ( t) Can wrte the backward dfferental equatons as P t = RP and ther soluton s ( ) ( 0 ) t t t = e R R P P = e snce P( 0) = I where n Rt n t e R n= 0 n! but ths computaton s not very effcent. We can also approxmate: e Rt = lm I+ R n t n n 1 t t R or e I R for large n n n Chapter 6 13
Continuous Time Markov Chain
Contnuous Tme Markov Chan Hu Jn Department of Electroncs and Communcaton Engneerng Hanyang Unversty ERICA Campus Contents Contnuous tme Markov Chan (CTMC) Propertes of sojourn tme Relatons Transton probablty
More information6. Stochastic processes (2)
6. Stochastc processes () Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 6. Stochastc processes () Contents Markov processes Brth-death processes 6. Stochastc processes () Markov process
More information6. Stochastic processes (2)
Contents Markov processes Brth-death processes Lect6.ppt S-38.45 - Introducton to Teletraffc Theory Sprng 5 Markov process Consder a contnuous-tme and dscrete-state stochastc process X(t) wth state space
More informationMarkov chains. Definition of a CTMC: [2, page 381] is a continuous time, discrete value random process such that for an infinitesimal
Markov chans M. Veeraraghavan; March 17, 2004 [Tp: Study the MC, QT, and Lttle s law lectures together: CTMC (MC lecture), M/M/1 queue (QT lecture), Lttle s law lecture (when dervng the mean response tme
More informationApplied Stochastic Processes
STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of
More informationConvergence of random processes
DS-GA 12 Lecture notes 6 Fall 216 Convergence of random processes 1 Introducton In these notes we study convergence of dscrete random processes. Ths allows to characterze phenomena such as the law of large
More informationReview: Discrete Event Random Processes. Hongwei Zhang
Revew: Dscrete Event Random Processes Hongwe Zhang http://www.cs.wayne.edu/~hzhang Outlne Markov chans and some renewal theory Markov chan Renewal processes, renewal reward processes, Markov renewal processes
More informationIntroduction to Continuous-Time Markov Chains and Queueing Theory
Introducton to Contnuous-Tme Markov Chans and Queueng Theory From DTMC to CTMC p p 1 p 12 1 2 k-1 k p k-1,k p k-1,k k+1 p 1 p 21 p k,k-1 p k,k-1 DTMC 1. Transtons at dscrete tme steps n=,1,2, 2. Past doesn
More informationTopics in Probability Theory and Stochastic Processes Steven R. Dunbar. Classes of States and Stationary Distributions
Steven R. Dunbar Department of Mathematcs 203 Avery Hall Unversty of Nebraska-Lncoln Lncoln, NE 68588-0130 http://www.math.unl.edu Voce: 402-472-3731 Fax: 402-472-8466 Topcs n Probablty Theory and Stochastc
More informationGoogle PageRank with Stochastic Matrix
Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d
More informationTCOM 501: Networking Theory & Fundamentals. Lecture 7 February 25, 2003 Prof. Yannis A. Korilis
TCOM 501: Networkng Theory & Fundamentals Lecture 7 February 25, 2003 Prof. Yanns A. Korls 1 7-2 Topcs Open Jackson Networks Network Flows State-Dependent Servce Rates Networks of Transmsson Lnes Klenrock
More informationEntropy of Markov Information Sources and Capacity of Discrete Input Constrained Channels (from Immink, Coding Techniques for Digital Recorders)
Entropy of Marov Informaton Sources and Capacty of Dscrete Input Constraned Channels (from Immn, Codng Technques for Dgtal Recorders). Entropy of Marov Chans We have already ntroduced the noton of entropy
More information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationEXPONENTIAL ERGODICITY FOR SINGLE-BIRTH PROCESSES
J. Appl. Prob. 4, 022 032 (2004) Prnted n Israel Appled Probablty Trust 2004 EXPONENTIAL ERGODICITY FOR SINGLE-BIRTH PROCESSES YONG-HUA MAO and YU-HUI ZHANG, Beng Normal Unversty Abstract An explct, computable,
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationRandomness and Computation
Randomness and Computaton or, Randomzed Algorthms Mary Cryan School of Informatcs Unversty of Ednburgh RC 208/9) Lecture 0 slde Balls n Bns m balls, n bns, and balls thrown unformly at random nto bns usually
More informationDynamic Programming. Lecture 13 (5/31/2017)
Dynamc Programmng Lecture 13 (5/31/2017) - A Forest Thnnng Example - Projected yeld (m3/ha) at age 20 as functon of acton taken at age 10 Age 10 Begnnng Volume Resdual Ten-year Volume volume thnned volume
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationHidden Markov Models & The Multivariate Gaussian (10/26/04)
CS281A/Stat241A: Statstcal Learnng Theory Hdden Markov Models & The Multvarate Gaussan (10/26/04) Lecturer: Mchael I. Jordan Scrbes: Jonathan W. Hu 1 Hdden Markov Models As a bref revew, hdden Markov models
More informationProblem Set 9 - Solutions Due: April 27, 2005
Problem Set - Solutons Due: Aprl 27, 2005. (a) Frst note that spam messages, nvtatons and other e-mal are all ndependent Posson processes, at rates pλ, qλ, and ( p q)λ. The event of the tme T at whch you
More informationON THE BURGERS EQUATION WITH A STOCHASTIC STEPPING STONE NOISY TERM
O THE BURGERS EQUATIO WITH A STOCHASTIC STEPPIG STOE OISY TERM Eaterna T. Kolovsa Comuncacón Técnca o I-2-14/11-7-22 PE/CIMAT On the Burgers Equaton wth a stochastc steppng-stone nosy term Eaterna T. Kolovsa
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationIntroduction to Hidden Markov Models
Introducton to Hdden Markov Models Alperen Degrmenc Ths document contans dervatons and algorthms for mplementng Hdden Markov Models. The content presented here s a collecton of my notes and personal nsghts
More informationAN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES. Rong-Rong Chen. ( University of Illinois at Urbana-Champaign)
AN EXTENDED CLASS OF TIME-CONTINUOUS BRANCHING PROCESSES Rong-Rong Chen ( Unversty of Illnos at Urbana-Champagn Abstract. Ths paper s devoted to studyng an extended class of tme-contnuous branchng processes,
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationSuggested solutions for the exam in SF2863 Systems Engineering. June 12,
Suggested solutons for the exam n SF2863 Systems Engneerng. June 12, 2012 14.00 19.00 Examner: Per Enqvst, phone: 790 62 98 1. We can thnk of the farm as a Jackson network. The strawberry feld s modelled
More informationFINITE-STATE MARKOV CHAINS
Chapter 4 FINITE-STATE MARKOV CHAINS 4.1 Introducton The countng processes {N(t), t 0} of Chapterss 2 and 3 have the property that N(t) changes at dscrete nstants of tme, but s defned for all real t 0.
More informationDS-GA 1002 Lecture notes 5 Fall Random processes
DS-GA Lecture notes 5 Fall 6 Introducton Random processes Random processes, also known as stochastc processes, allow us to model quanttes that evolve n tme (or space n an uncertan way: the trajectory of
More informationComplex Numbers, Signals, and Circuits
Complex Numbers, Sgnals, and Crcuts 3 August, 009 Complex Numbers: a Revew Suppose we have a complex number z = x jy. To convert to polar form, we need to know the magntude of z and the phase of z. z =
More informationHidden Markov Models
Hdden Markov Models Namrata Vaswan, Iowa State Unversty Aprl 24, 204 Hdden Markov Model Defntons and Examples Defntons:. A hdden Markov model (HMM) refers to a set of hdden states X 0, X,..., X t,...,
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationCase Study of Markov Chains Ray-Knight Compactification
Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationStrong Markov property: Same assertion holds for stopping times τ.
Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up
More informationQueuing system theory
Elements of queung system: Queung system theory Every queung system conssts of three elements: An arrval process: s characterzed by the dstrbuton of tme between the arrval of successve customers, the mean
More informationDynamic Programming. Preview. Dynamic Programming. Dynamic Programming. Dynamic Programming (Example: Fibonacci Sequence)
/24/27 Prevew Fbonacc Sequence Longest Common Subsequence Dynamc programmng s a method for solvng complex problems by breakng them down nto smpler sub-problems. It s applcable to problems exhbtng the propertes
More informationCS 3750 Machine Learning Lecture 6. Monte Carlo methods. CS 3750 Advanced Machine Learning. Markov chain Monte Carlo
CS 3750 Machne Learnng Lectre 6 Monte Carlo methods Mlos Haskrecht mlos@cs.ptt.ed 5329 Sennott Sqare Markov chan Monte Carlo Importance samplng: samples are generated accordng to Q and every sample from
More informationChapter 4: Root Finding
Chapter 4: Root Fndng Startng values Closed nterval methods (roots are search wthn an nterval o Bsecton Open methods (no nterval o Fxed Pont o Newton-Raphson o Secant Method Repeated roots Zeros of Hgher-Dmensonal
More informationCSCE 790S Background Results
CSCE 790S Background Results Stephen A. Fenner September 8, 011 Abstract These results are background to the course CSCE 790S/CSCE 790B, Quantum Computaton and Informaton (Sprng 007 and Fall 011). Each
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More informationFluctuation Results For Quadratic Continuous-State Branching Process
IOSR Journal of Mathematcs (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 3 Ver. III (May - June 2017), PP 54-61 www.osrjournals.org Fluctuaton Results For Quadratc Contnuous-State Branchng
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationRandom Walks on Digraphs
Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected
More informationTHE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationDynamic Systems on Graphs
Prepared by F.L. Lews Updated: Saturday, February 06, 200 Dynamc Systems on Graphs Control Graphs and Consensus A network s a set of nodes that collaborates to acheve what each cannot acheve alone. A network,
More informationAnalysis of Discrete Time Queues (Section 4.6)
Analyss of Dscrete Tme Queues (Secton 4.6) Copyrght 2002, Sanjay K. Bose Tme axs dvded nto slots slot slot boundares Arrvals can only occur at slot boundares Servce to a job can only start at a slot boundary
More informationSolutions to Problem Set 6
Solutons to Problem Set 6 Problem 6. (Resdue theory) a) Problem 4.7.7 Boas. n ths problem we wll solve ths ntegral: x sn x x + 4x + 5 dx: To solve ths usng the resdue theorem, we study ths complex ntegral:
More informationNetwork of Markovian Queues. Lecture
etwork of Markovan Queues etwork of Markovan Queues ETW09 20 etwork queue ed, G E ETW09 20 λ If the frst queue was not empty Then the tme tll the next arrval to the second queue wll be equal to the servce
More information( ) ( ) ( ) ( ) STOCHASTIC SIMULATION FOR BLOCKED DATA. Monte Carlo simulation Rejection sampling Importance sampling Markov chain Monte Carlo
SOCHASIC SIMULAIO FOR BLOCKED DAA Stochastc System Analyss and Bayesan Model Updatng Monte Carlo smulaton Rejecton samplng Importance samplng Markov chan Monte Carlo Monte Carlo smulaton Introducton: If
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationProblem Solving in Math (Math 43900) Fall 2013
Problem Solvng n Math (Math 43900) Fall 2013 Week four (September 17) solutons Instructor: Davd Galvn 1. Let a and b be two nteger for whch a b s dvsble by 3. Prove that a 3 b 3 s dvsble by 9. Soluton:
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationRate of Absorption and Stimulated Emission
MIT Department of Chemstry 5.74, Sprng 005: Introductory Quantum Mechancs II Instructor: Professor Andre Tokmakoff p. 81 Rate of Absorpton and Stmulated Emsson The rate of absorpton nduced by the feld
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationQueueing Networks II Network Performance
Queueng Networks II Network Performance Davd Tpper Assocate Professor Graduate Telecommuncatons and Networkng Program Unversty of Pttsburgh Sldes 6 Networks of Queues Many communcaton systems must be modeled
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationLecture 3: Shannon s Theorem
CSE 533: Error-Correctng Codes (Autumn 006 Lecture 3: Shannon s Theorem October 9, 006 Lecturer: Venkatesan Guruswam Scrbe: Wdad Machmouch 1 Communcaton Model The communcaton model we are usng conssts
More informationHidden Markov Model Cheat Sheet
Hdden Markov Model Cheat Sheet (GIT ID: dc2f391536d67ed5847290d5250d4baae103487e) Ths document s a cheat sheet on Hdden Markov Models (HMMs). It resembles lecture notes, excet that t cuts to the chase
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationCS-433: Simulation and Modeling Modeling and Probability Review
CS-433: Smulaton and Modelng Modelng and Probablty Revew Exercse 1. (Probablty of Smple Events) Exercse 1.1 The owner of a camera shop receves a shpment of fve cameras from a camera manufacturer. Unknown
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationChanging Topology and Communication Delays
Prepared by F.L. Lews Updated: Saturday, February 3, 00 Changng Topology and Communcaton Delays Changng Topology The graph connectvty or topology may change over tme. Let G { G, G,, G M } wth M fnte be
More informationBernoulli Numbers and Polynomials
Bernoull Numbers and Polynomals T. Muthukumar tmk@tk.ac.n 17 Jun 2014 The sum of frst n natural numbers 1, 2, 3,..., n s n n(n + 1 S 1 (n := m = = n2 2 2 + n 2. Ths formula can be derved by notng that
More informationOn complexity and randomness of Markov-chain prediction
On complexty and randomness of Markov-chan predcton Joel Ratsaby Department of Electrcal and Electroncs Engneerng Arel Unversty Arel, ISRAEL Emal: ratsaby@arelacl Abstract Let {X t : t Z} be a sequence
More informationMARKOV CHAINS. 5. Recurrence and transience. and the length of the rth excursion to i by. Part IB Michaelmas 2009 YMS.
Part IB Mchaelmas 009 YMS MARKOV CHAINS E-mal: yms@statslab.cam.ac.u 5. Recurrence transence Recurrence transence; equvalence of transence summablty of n- step transton probabltes; equvalence of recurrence
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationPrimer on High-Order Moment Estimators
Prmer on Hgh-Order Moment Estmators Ton M. Whted July 2007 The Errors-n-Varables Model We wll start wth the classcal EIV for one msmeasured regressor. The general case s n Erckson and Whted Econometrc
More informationLecture 17 : Stochastic Processes II
: Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationMarginal Effects in Probit Models: Interpretation and Testing. 1. Interpreting Probit Coefficients
ECON 5 -- NOE 15 Margnal Effects n Probt Models: Interpretaton and estng hs note ntroduces you to the two types of margnal effects n probt models: margnal ndex effects, and margnal probablty effects. It
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationDesigning of Combined Continuous Lot By Lot Acceptance Sampling Plan
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 709 Desgnng o Combned Contnuous Lot By Lot Acceptance Samplng Plan S. Subhalakshm 1 Dr. S. Muthulakshm 2 1 Research Scholar,
More informationProbability and Random Variable Primer
B. Maddah ENMG 622 Smulaton 2/22/ Probablty and Random Varable Prmer Sample space and Events Suppose that an eperment wth an uncertan outcome s performed (e.g., rollng a de). Whle the outcome of the eperment
More informationHopfield networks and Boltzmann machines. Geoffrey Hinton et al. Presented by Tambet Matiisen
Hopfeld networks and Boltzmann machnes Geoffrey Hnton et al. Presented by Tambet Matsen 18.11.2014 Hopfeld network Bnary unts Symmetrcal connectons http://www.nnwj.de/hopfeld-net.html Energy functon The
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationA PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS
HCMC Unversty of Pedagogy Thong Nguyen Huu et al. A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS Thong Nguyen Huu and Hao Tran Van Department of mathematcs-nformaton,
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 3: Large deviations bounds and applications Lecturer: Sanjeev Arora
prnceton unv. F 13 cos 521: Advanced Algorthm Desgn Lecture 3: Large devatons bounds and applcatons Lecturer: Sanjeev Arora Scrbe: Today s topc s devaton bounds: what s the probablty that a random varable
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationFrom Biot-Savart Law to Divergence of B (1)
From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to
More informationErratum: A Generalized Path Integral Control Approach to Reinforcement Learning
Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of
More informationDigital Modems. Lecture 2
Dgtal Modems Lecture Revew We have shown that both Bayes and eyman/pearson crtera are based on the Lkelhood Rato Test (LRT) Λ ( r ) < > η Λ r s called observaton transformaton or suffcent statstc The crtera
More informationLecture 6 More on Complete Randomized Block Design (RBD)
Lecture 6 More on Complete Randomzed Block Desgn (RBD) Multple test Multple test The multple comparsons or multple testng problem occurs when one consders a set of statstcal nferences smultaneously. For
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationCS294 Topics in Algorithmic Game Theory October 11, Lecture 7
CS294 Topcs n Algorthmc Game Theory October 11, 2011 Lecture 7 Lecturer: Chrstos Papadmtrou Scrbe: Wald Krchene, Vjay Kamble 1 Exchange economy We consder an exchange market wth m agents and n goods. Agent
More informationMultidimensional Random Motion with Uniformly Distributed Changes of Direction and Erlang Steps
Multdmensonal Random Moton wth Unformly Dstrbuted Changes of Drecton and Erlang Steps Anatoly A. Pogoru a,, Ramón M. Rodríguez-Dagnno b,, a Department of Mathematcs, Zhytomyr State Unversty. b Electrcal
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More information