Stochastic processes are collections of random variables indexed by time or space that are used to model randomly varying phenomena. This document introduces key concepts of stochastic processes including:
1. Stochastic processes can be discrete-time or continuous-time depending on whether the index set is discrete or continuous.
2. Finite-dimensional distributions characterize stochastic processes statistically by describing the joint distributions of subsets of the random variables.
3. Stationary processes have statistics like mean and covariance that do not depend on absolute time. Gaussian and Poisson processes are important stationary processes.
This document discusses deep Kalman filters, which combine deep learning and Kalman filtering. It proposes replacing the linear transformations in classical Kalman filters with nonlinear transformations parameterized by neural networks. This allows the model to learn patterns in noisy sequential data and model the effects of external actions. The model is evaluated on synthetic and real patient data, showing it can successfully perform counterfactual inference about the effects of anti-diabetic drugs on diabetic patients.
1) The document outlines the key steps and equations of the Kalman filter algorithm for optimal state estimation.
2) It describes the Kalman filter as a recursive algorithm that uses a system's dynamics model and noisy measurements to produce optimal estimates of unknown variables.
3) The algorithm involves two main steps - prediction using the system model to produce an estimate, and correction using new measurements to update the estimate.
time domain analysis, Rise Time, Delay time, Damping Ratio, Overshoot, Settli...Waqas Afzal
Time Response- Transient, Steady State
Standard Test Signals- U(t), S(t), R(t)
Analysis of First order system - for Step input
Analysis of second order system -for Step input
Time Response Specifications- Rise Time, Delay time, Damping Ratio, Overshoot, Settling Time
Calculations
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
This document provides a summary of key concepts in nonlinear systems and control theory that are necessary background for subsequent chapters. It introduces notation used throughout the book and defines stability concepts such as Lyapunov stability, asymptotic stability, and exponential stability. It also summarizes Lyapunov's direct method, which allows determining stability properties of an equilibrium point from the properties of the system function f(x) and its relationship to a positive definite function V(x).
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
I am Stuart M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master in Statistics from, the University of Greenwich, UK. I have been helping students with their Assignments for the past 6 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
This document discusses deep Kalman filters, which combine deep learning and Kalman filtering. It proposes replacing the linear transformations in classical Kalman filters with nonlinear transformations parameterized by neural networks. This allows the model to learn patterns in noisy sequential data and model the effects of external actions. The model is evaluated on synthetic and real patient data, showing it can successfully perform counterfactual inference about the effects of anti-diabetic drugs on diabetic patients.
1) The document outlines the key steps and equations of the Kalman filter algorithm for optimal state estimation.
2) It describes the Kalman filter as a recursive algorithm that uses a system's dynamics model and noisy measurements to produce optimal estimates of unknown variables.
3) The algorithm involves two main steps - prediction using the system model to produce an estimate, and correction using new measurements to update the estimate.
time domain analysis, Rise Time, Delay time, Damping Ratio, Overshoot, Settli...Waqas Afzal
Time Response- Transient, Steady State
Standard Test Signals- U(t), S(t), R(t)
Analysis of First order system - for Step input
Analysis of second order system -for Step input
Time Response Specifications- Rise Time, Delay time, Damping Ratio, Overshoot, Settling Time
Calculations
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
This document provides a summary of key concepts in nonlinear systems and control theory that are necessary background for subsequent chapters. It introduces notation used throughout the book and defines stability concepts such as Lyapunov stability, asymptotic stability, and exponential stability. It also summarizes Lyapunov's direct method, which allows determining stability properties of an equilibrium point from the properties of the system function f(x) and its relationship to a positive definite function V(x).
This document discusses dynamical systems. It defines a dynamical system as a system that changes over time according to fixed rules determining how its state changes from one time to the next. It then covers:
- The two parts of a dynamical system: state space and function determining next state.
- Classification of systems as deterministic/stochastic, discrete/continuous, linear/nonlinear, and autonomous/nonautonomous.
- Examples of discrete and continuous models, differential equations, and linear vs nonlinear systems.
- Terminology including phase space, phase curve, phase portrait, and attractors.
- Analysis methods including fixed points, stability, and perturbation analysis.
- Examples of harmonic oscillator,
I am Stuart M. I am a Statistics Assignment Expert at statisticsassignmenthelp.com. I hold a Master in Statistics from, the University of Greenwich, UK. I have been helping students with their Assignments for the past 6 years. I solve assignments related to Statistics.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
The feedback-control-for-distributed-systemsCemal Ardil
The document summarizes a study on feedback control synthesis for distributed systems. The study proposes a zone control approach, where the state space is partitioned into zones defined by observable points. Control actions are piecewise constant functions that only change when the system transitions between zones. An optimization problem is formulated to determine the optimal constant control value for each zone. Gradient formulas are derived to solve this using numerical optimization methods. The zone control approach was tested on heat exchanger process control problems and showed more robust performance than alternative methods.
This document provides an overview of key concepts related to random processes, including:
1) Random processes can be described statistically using properties like the mean, variance, autocorrelation, and spectral density.
2) Stationary and ergodic processes have statistical properties that do not vary over time or between samples.
3) Autocorrelation describes the dependency of a process on past values, while spectral density describes the frequency content.
4) Cross-correlation and cross-spectral density characterize the relationship between two random processes.
5) The response of a linear system to a random input can be related to the input's spectral density using the system's transfer function.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
1) Delayed renewal processes occur when the first renewal time has a different distribution than subsequent renewal times. The fundamental theorems state that the average number of renewals divided by time approaches the reciprocal of the expected subsequent renewal time.
2) A stationary renewal process models a process that has been ongoing infinitely far in the past, so the first renewal time has the limiting excess lifetime distribution of a standard renewal process. Properties of these processes do not depend on time.
3) Cumulative processes associate a random value with each renewal interval. The fundamental theorems state that the average accumulated value by time t approaches the expected value divided by the expected renewal time.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Skiena algorithm 2007 lecture09 linear sortingzukun
- The document discusses linear sorting algorithms like quicksort.
- It provides pseudocode for quicksort and explains its best, average, and worst case time complexities. Quicksort runs in O(n log n) time on average but can be O(n^2) in the worst case if the pivot element is selected poorly.
- Randomized quicksort is discussed as a way to achieve expected O(n log n) time for any input by selecting the pivot randomly.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
The document discusses concepts related to linear time-invariant systems represented by state-space models. It provides:
1) An overview of desired learning outcomes related to solving state equations and understanding controllability and observability.
2) Details on solving the homogeneous and non-homogeneous state equations, including definitions of zero-state and zero-input responses.
3) The condition for a system to be completely state controllable, which is that the controllability matrix must be of full rank.
Quicksort has a worst case time complexity of O(n^2) when the pivot is always the minimum or maximum element, dividing the array into arrays of size 1 and n-1 on each iteration. The best case of O(n log n) occurs when the pivot evenly divides the array into two halves. On average, quicksort runs in O(n log n) time. Practical implementations avoid the worst case by randomly selecting the pivot or choosing the median element as the pivot on each iteration.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
Stochastic Optimal Control & Information Theoretic DualitiesHaruki Nishimura
In this series of two presentations we discuss two different, yet related, approaches to general nonlinear stochastic control: stochastic optimal control and information theoretic control. This week is the first half and the main focus is on the stochastic optimal control. We begin our discussion by reviewing the deterministic case and see that Bellman's principle of optimality leads to the Hamilton-Jacobi-Bellman equation. We then learn how the problem can be extended to handle stochasticity in the system dynamics, where the Wiener noise affects the resulting value function. Difficulties around solving the stochastic dynamic program are presented, leading to the information theoretic control that is based on another notion of optimality.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
1. The document discusses discrete-time control systems, where variables can only change at discrete time intervals denoted by kT.
2. It explains that continuous signals must be sampled and quantified before being processed digitally. This involves sample-and-hold circuits and analog-to-digital converters.
3. Discrete-time control systems are classified based on whether the process is continuous or discrete. The most common are discrete-time controllers for continuous processes, which require discretizing analog signals.
This document provides a summary of statistical linearization methodologies for analyzing the response of nonlinear structures subjected to seismic excitation. It discusses several statistical linearization techniques for both elastic and inelastic systems, including the standard second-order method for systems without hysteresis and stochastic averaging for hysteretic systems. It also presents an example framework that uses statistical linearization and response spectra to analyze vibro-impact systems subjected to earthquake ground motions.
An1 derivat.ro fizica-2_quantum physics 2 2010 2011_24125Robin Cruise Jr.
1. The document discusses key concepts in quantum physics including wave-particle duality, quantum states as linear combinations or superpositions, and the uncertainty principle.
2. It outlines the quantum postulates including that states are elements of a state space, physical quantities are represented by operators with real eigenvalues, and measurements change the system state.
3. It introduces the Schrodinger equation which describes how quantum states evolve over time as governed by the Hamiltonian operator. Specific examples are provided including the rectangular potential barrier and quantum harmonic oscillator.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Este estudo analisa pesquisas nacionais e internacionais sobre a linguagem XBRL entre 2009-2013, comparando os temas, autores e periódicos mais relevantes. Os estudos brasileiros focam no uso do XBRL no setor público, enquanto os internacionais abordam sua adoção e difusão e a asseguração de documentos XBRL. O periódico com mais artigos é o International Journal of Accounting Information e o autor mais citado é Roger Debreceny.
Este documento discute as relações entre cultura e educação e as políticas culturais. A cultura deve ser vista como um movimento em mão dupla entre a cidade e o interior, preservando as manifestações culturais locais e promovendo o acesso à cultura erudita. As políticas culturais devem equilibrar a conservação do patrimônio e a promoção da criação artística. A cultura não deve ser vista apenas como expressão elitista, mas como modo de vida e produção das comunidades.
El periódico de hoy informa sobre varios temas importantes. Se discuten las negociaciones comerciales entre los países y los esfuerzos en curso para llegar a un acuerdo. También se informa sobre una elección local y los candidatos que compiten por un puesto político. Finalmente, hay un artículo de opinión sobre el cambio climático y las acciones que los gobiernos deben tomar para abordar este problema global.
The feedback-control-for-distributed-systemsCemal Ardil
The document summarizes a study on feedback control synthesis for distributed systems. The study proposes a zone control approach, where the state space is partitioned into zones defined by observable points. Control actions are piecewise constant functions that only change when the system transitions between zones. An optimization problem is formulated to determine the optimal constant control value for each zone. Gradient formulas are derived to solve this using numerical optimization methods. The zone control approach was tested on heat exchanger process control problems and showed more robust performance than alternative methods.
This document provides an overview of key concepts related to random processes, including:
1) Random processes can be described statistically using properties like the mean, variance, autocorrelation, and spectral density.
2) Stationary and ergodic processes have statistical properties that do not vary over time or between samples.
3) Autocorrelation describes the dependency of a process on past values, while spectral density describes the frequency content.
4) Cross-correlation and cross-spectral density characterize the relationship between two random processes.
5) The response of a linear system to a random input can be related to the input's spectral density using the system's transfer function.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
1) Delayed renewal processes occur when the first renewal time has a different distribution than subsequent renewal times. The fundamental theorems state that the average number of renewals divided by time approaches the reciprocal of the expected subsequent renewal time.
2) A stationary renewal process models a process that has been ongoing infinitely far in the past, so the first renewal time has the limiting excess lifetime distribution of a standard renewal process. Properties of these processes do not depend on time.
3) Cumulative processes associate a random value with each renewal interval. The fundamental theorems state that the average accumulated value by time t approaches the expected value divided by the expected renewal time.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
Skiena algorithm 2007 lecture09 linear sortingzukun
- The document discusses linear sorting algorithms like quicksort.
- It provides pseudocode for quicksort and explains its best, average, and worst case time complexities. Quicksort runs in O(n log n) time on average but can be O(n^2) in the worst case if the pivot element is selected poorly.
- Randomized quicksort is discussed as a way to achieve expected O(n log n) time for any input by selecting the pivot randomly.
My talk in the MCQMC Conference 2016, Stanford University. The talk is about Multilevel Hybrid Split Step Implicit Tau-Leap
for Stochastic Reaction Networks.
The document discusses concepts related to linear time-invariant systems represented by state-space models. It provides:
1) An overview of desired learning outcomes related to solving state equations and understanding controllability and observability.
2) Details on solving the homogeneous and non-homogeneous state equations, including definitions of zero-state and zero-input responses.
3) The condition for a system to be completely state controllable, which is that the controllability matrix must be of full rank.
Quicksort has a worst case time complexity of O(n^2) when the pivot is always the minimum or maximum element, dividing the array into arrays of size 1 and n-1 on each iteration. The best case of O(n log n) occurs when the pivot evenly divides the array into two halves. On average, quicksort runs in O(n log n) time. Practical implementations avoid the worst case by randomly selecting the pivot or choosing the median element as the pivot on each iteration.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
Stochastic Optimal Control & Information Theoretic DualitiesHaruki Nishimura
In this series of two presentations we discuss two different, yet related, approaches to general nonlinear stochastic control: stochastic optimal control and information theoretic control. This week is the first half and the main focus is on the stochastic optimal control. We begin our discussion by reviewing the deterministic case and see that Bellman's principle of optimality leads to the Hamilton-Jacobi-Bellman equation. We then learn how the problem can be extended to handle stochasticity in the system dynamics, where the Wiener noise affects the resulting value function. Difficulties around solving the stochastic dynamic program are presented, leading to the information theoretic control that is based on another notion of optimality.
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
1. The document discusses discrete-time control systems, where variables can only change at discrete time intervals denoted by kT.
2. It explains that continuous signals must be sampled and quantified before being processed digitally. This involves sample-and-hold circuits and analog-to-digital converters.
3. Discrete-time control systems are classified based on whether the process is continuous or discrete. The most common are discrete-time controllers for continuous processes, which require discretizing analog signals.
This document provides a summary of statistical linearization methodologies for analyzing the response of nonlinear structures subjected to seismic excitation. It discusses several statistical linearization techniques for both elastic and inelastic systems, including the standard second-order method for systems without hysteresis and stochastic averaging for hysteretic systems. It also presents an example framework that uses statistical linearization and response spectra to analyze vibro-impact systems subjected to earthquake ground motions.
An1 derivat.ro fizica-2_quantum physics 2 2010 2011_24125Robin Cruise Jr.
1. The document discusses key concepts in quantum physics including wave-particle duality, quantum states as linear combinations or superpositions, and the uncertainty principle.
2. It outlines the quantum postulates including that states are elements of a state space, physical quantities are represented by operators with real eigenvalues, and measurements change the system state.
3. It introduces the Schrodinger equation which describes how quantum states evolve over time as governed by the Hamiltonian operator. Specific examples are provided including the rectangular potential barrier and quantum harmonic oscillator.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Este estudo analisa pesquisas nacionais e internacionais sobre a linguagem XBRL entre 2009-2013, comparando os temas, autores e periódicos mais relevantes. Os estudos brasileiros focam no uso do XBRL no setor público, enquanto os internacionais abordam sua adoção e difusão e a asseguração de documentos XBRL. O periódico com mais artigos é o International Journal of Accounting Information e o autor mais citado é Roger Debreceny.
Este documento discute as relações entre cultura e educação e as políticas culturais. A cultura deve ser vista como um movimento em mão dupla entre a cidade e o interior, preservando as manifestações culturais locais e promovendo o acesso à cultura erudita. As políticas culturais devem equilibrar a conservação do patrimônio e a promoção da criação artística. A cultura não deve ser vista apenas como expressão elitista, mas como modo de vida e produção das comunidades.
El periódico de hoy informa sobre varios temas importantes. Se discuten las negociaciones comerciales entre los países y los esfuerzos en curso para llegar a un acuerdo. También se informa sobre una elección local y los candidatos que compiten por un puesto político. Finalmente, hay un artículo de opinión sobre el cambio climático y las acciones que los gobiernos deben tomar para abordar este problema global.
Хозяйственный риск: понятие и научные подходыITMO University
В данной статье приведен анализ истории научных исследований проблем хозяйственного риска; определены научные подходы исследования рисков, изучены понятия «риск» и «хозяйственный риск»
Los proyectos son iniciativas que requieren planificación y ejecución para lograr un objetivo específico. Se necesita definir claramente el alcance, los recursos, el cronograma y las responsabilidades para garantizar el éxito del proyecto. Un enfoque de gestión de proyectos efectivo incluye monitorear el progreso, administrar los riesgos y asegurar que el proyecto se complete a tiempo y dentro del presupuesto.
So you've ready the Sprint book (or heard about Google Design Sprints) but you're trying to figure out how to actually do that? We'll give you a quick overview along with our tips (note this is much better in person, so reach out if you'd like us to come give a talk).
A risk assessment was conducted for a film shoot in the Green Screen Room at De Aston School on 07/12/15. Hazards identified included cables, a tripod, and a table leg that could pose trip hazards. Measures taken to prevent injury included neatly securing cables to the floor with bright tape, pointing out the tripod legs to cast and crew, and pulling a table forward during shots. Responsible parties were assigned to each hazard identified. The overall risks were determined to be adequately addressed and given a green "G" rating.
Danyell Spring has over 15 years of experience in strategic business planning, data analysis, financial modeling, and process improvement. She currently works as a Pharmacy Analytics Manager at Providence Health & Services, where she develops analytical approaches to identify over $20M in annual cost savings opportunities. Previously, she held roles as a Program Manager in decision support, Sr. Financial Analyst, and Financial Analyst focused on contract analysis, forecasting, and cost savings identification. She has an MBA from Baker University and a BS in Accounting from the University of Phoenix.
ОПТИМИЗАЦИЯ КЛАСТЕРА С ОГРАНИЧЕННОЙ ДОСТУПНОСТЬЮ КЛАСТЕРНЫХ ГРУППITMO University
Определены состав и число (кратность резервирования) кластерных групп различной функциональной комплектации, обеспечивающие минимальную стоимость реализации системы при заданных требованиях по ее надежности.
La expresión oral es una habilidad importante para la comunicación. Es necesario practicarla regularmente para mejorar la fluidez, la claridad de ideas y la capacidad de transmitir mensajes de manera efectiva. Una buena expresión oral requiere preparación, confianza en uno mismo y escuchar activamente a los demás.
El documento describe el origen y naturaleza de las nueve Musas de la mitología griega. Zeus engendró a las nueve Musas con Mnemósine en el monte Helicón. Cada Musa preside sobre un arte o ciencia particular como la poesía épica, la historia, la música, la tragedia, etc. Se enumeran los nombres de las nueve Musas y sus dominios respectivos.
1. The document discusses stochastic processes and their properties. A stochastic process is a collection of random variables indexed by time or some other parameter.
2. There are two main types of stationarity discussed - strict-sense stationarity and wide-sense stationarity. A process is strictly stationary if its joint probability distributions do not change when the time indices are shifted by a constant. A process is widely stationary if its mean is constant over time and its autocorrelation depends only on the difference between time indices.
3. For Gaussian processes, wide-sense stationarity is equivalent to strict-sense stationarity since the joint distributions of Gaussian variables are determined completely by their means and autocorrelations.
lecture about different methods and explainanation.pptsadafshahbaz7777
1. The document discusses stochastic processes and their properties. A stochastic process is a collection of random variables indexed by time or some other parameter.
2. There are two main types of stationarity discussed - strict-sense stationarity and wide-sense stationarity. A process is strictly stationary if its joint probability distributions do not change when the time indices are shifted by a constant. A process is widely stationary if its mean is constant over time and its autocorrelation depends only on the difference between time indices.
3. For Gaussian processes, wide-sense stationarity is equivalent to strict-sense stationarity since the joint distributions of Gaussian variables are determined completely by their means and autocorrelations.
1. The document provides a list of 2 mark questions and answers related to the Signals and Systems subject for the 3rd semester IT students.
2. It includes definitions of key terms like signal, system, different types of signals and their classifications. Properties of Fourier series and Fourier transforms are also covered.
3. The questions address topics ranging from periodic/aperiodic signals, even/odd signals, unit step and impulse functions, Fourier series, Fourier transforms, Laplace transforms, linear and time invariant systems.
The document provides an overview and history of the wavelet transform. It can be summarized as follows:
1. The wavelet transform was developed to address limitations of the Fourier transform and short-time Fourier transform in analyzing signals both in time and frequency. It uses wavelets of limited duration that can be scaled and translated.
2. The history of the wavelet transform began in 1909 with Haar wavelets. The concept of wavelets was then proposed in 1981 and the term was coined in 1984. Important developments included the construction of additional orthogonal wavelets in 1985, the proposal of the multiresolution concept in 1988, and the fast wavelet transform algorithm in 1989, enabling numerous applications.
3.
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Part I. Read Sections #29.5 (Long-run properties of Markov Chains), #29.6 (First Passage
Times) of the attached, and write a summary report.
Note that the summary report has to be prepared on a word processor (e.g., MS Word), and it has
to be submitted through our class canvas system. Your report will be formatted with the
following traits:
The title page should include course title, student name, and the date.
There is no page limit but the article summary should be at least 2 pages long, single spaced
throughout.
Use a standard font (Times New Roman 12).
Use 1 inch margins for top, bottom, left, and right.
Use proper punctuation, spelling, and grammar.
All pages (with the exception of the title page) should be numbered.
1
29C H A P T E R
Markov Chains
Chapter 16 focused on decision making in the face of uncertainty about one futureevent (learning the true state of nature). However, some decisions need to take into
account uncertainty about many future events. We now begin laying the groundwork for
decision making in this broader context.
In particular, this chapter presents probability models for processes that evolve over
time in a probabilistic manner. Such processes are called stochastic processes. After briefly
introducing general stochastic processes in the first section, the remainder of the chapter
focuses on a special kind called a Markov chain. Markov chains have the special prop-
erty that probabilities involving how the process will evolve in the future depend only on
the present state of the process, and so are independent of events in the past. Many
processes fit this description, so Markov chains provide an especially important kind of
probability model.
For example, Chap. 17 mentioned that continuous-time Markov chains (described in
Sec. 29.8) are used to formulate most of the basic models of queueing theory. Markov
chains also provided the foundation for the study of Markov decision models in Chap. 19.
There are a wide variety of other applications of Markov chains as well. A considerable
number of books and articles present some of these applications. One is Selected Refer-
ence 4, which describes applications in such diverse areas as the classification of
customers, DNA sequencing, the analysis of genetic networks, the estimation of sales
demand over time, and credit rating. Selected Reference 6 focuses on applications in fi-
nance and Selected Reference 3 describes applications for analyzing baseball strategy.
The list goes on and on, but let us turn now to a description of stochastic processes in
general and Markov chains in particular.
■ 29.1 STOCHASTIC PROCESSES
A stochastic process is defined as an indexed collection of random variables {Xt},
where the index t runs through a given set T. Often T is taken to be the set of non-
negative ...
The document provides notes on signals and systems from an EECE 301 course. It includes:
- An overview of continuous-time (C-T) and discrete-time (D-T) signal and system models.
- Details on chapters covering differentials/differences, convolution, Fourier analysis (both C-T and D-T), Laplace transforms, and Z-transforms.
- Examples of calculating the Fourier transform of specific signals like a decaying exponential and rectangular pulse. These illustrate properties of the Fourier transform.
The document discusses random phenomena and random processes. Some key points:
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- A random process is the collection of all possible time histories that could result from random phenomena. Individual time histories are called sample functions.
- Random processes can be described using averages over ensembles of sample functions or over time from a single sample function. Stationary and ergodic processes allow the use of time averages.
- Random variables, power spectral densities, and probability distributions provide information about random processes and allow their characterization in different domains.
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1) Stochastic processes are sequences of random variables indexed by time that evolve randomly over time. The value at each time Xt may depend on previous values.
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3) Autocovariance and autocorrelation functions describe the covariance and correlation between values at different times and are important tools for analyzing stationary processes.
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International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document summarizes key concepts related to martingales and stopping times in probability theory. It begins with an introduction to martingales and their properties. A martingale is a stochastic process where the expected future value is equal to the present value. The document then discusses stopping times, which are random variables that determine when a stochastic process will stop. A key theorem discussed is Doob's Optional Stopping Theorem, which establishes conditions under which a martingale that is stopped at a random time remains a martingale. The document concludes by defining stochastic processes and stopped processes, which are processes that are stopped at a specified stopping time.
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1) The document discusses using the residue theorem to evaluate a complex contour integral to calculate the Laplace transform of the output of an ideal sampler. This provides a closed-form solution that is less painful than the infinite series form.
2) An ideal sampler can be modeled as a carrier signal modulated by the input signal. The output of the sampler is then sent to a zero-order hold.
3) By choosing an appropriate contour, the complex contour integral can be evaluated using the residue theorem. This gives the Laplace transform of the sampler output in terms of the residues of the integrand's poles.
The Fourier transform relates a signal in the time domain, x(t), to its frequency domain representation, X(jw). It represents the frequency content of the signal. The Fourier transform is a linear operation, and time shifts in the time domain result in phase shifts in the frequency domain. Differentiation in the time domain corresponds to multiplication by jw in the frequency domain. Convolution becomes simple multiplication in the frequency domain. These properties allow differential equations and systems with convolution to be solved using algebraic operations by working in the frequency domain.
3. Frequency-Domain Analysis of Continuous-Time Signals and Systems.pdfTsegaTeklewold1
This document discusses frequency-domain analysis of continuous-time signals and systems. It defines the continuous-time Fourier series and Fourier transform, which decompose continuous-time signals into a set of continuous elementary signals. Properties of the continuous-time Fourier series such as linearity, differentiation, shifting, scaling, and convolution are described. The document also discusses the convergence conditions and properties of the continuous-time Fourier transform.
1. The figure shows an electrical circuit driven by a heartbeat generator. Its output is associated with a recorder for later examination. The document discusses Fourier analysis of periodic and aperiodic signals from the circuit.
2. The document discusses Fourier analysis properties such as linearity, time shifting, differentiation, and integration that are applied to analyze signals from various systems like the stock market or a microphone.
3. The document discusses using Fourier analysis to transform voltage level signals from a microphone into sound waves for recording and communication. It also discusses properties of the continuous-time Fourier series such as linearity and time shifting that are applied to analyze the signals.
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2. 56 2 Introduction to Stochastic Processes
2.2 Finite-Dimensional Distributions of Stochastic Processes
A stochastic process fXt ; t 2 T g can be characterized in a statistical sense by its
finite-dimensional distributions.
Definition 2.2. The finite-dimensional distributions of a stochastic process
fXt ; t 2 T g are defined by the family of all joint distribution functions
Ft1 ;:::;tn .x1 ; : : : ; xn / D P .Xt1 < x1 ; : : : ; Xtn < xn /;
where n D 1; 2; : : : and t1 ; : : : ; tn 2 T .
The family of introduced distribution functions
F D fFt1 ;:::;tn ; t1 ; : : : ; tn 2 T ; n D 1; 2; : : :g
satisfies the following, specified consistency conditions:
(a) For all positive integers n, m and indices t1 ; : : : ; tnCm 2 T
lim ::: lim Ft1 ;:::;tn ;tnC1 ;:::;tnCm .x1 ; : : : ; xn ; xnC1 ; : : : ; xnCm /
xnC1 !1 xnCm !1
D Ft1 ;:::;tn .x1 ; : : : ; xn /; x1 ; : : : ; xn 2 R :
(b) For all permutations .i1 ; : : : ; in / of the numbers f1; 2; : : : ; ng
Fs1 ;:::;sn .xi1 ; : : : ; xin / D Ft1 ;:::;tn .x1 ; : : : ; xn /; x1 ; : : : ; xn 2 R ;
where sj D tij ; j D 1; : : : ; n:
Definition 2.3. If the family F of joint distribution functions defined previously
satisfies conditions (a) and (b), then we say that F satisfies the consistency
conditions.
The following theorem is a basic one in probability theory and ensures the
existence of a stochastic process (in general of a collection of random variables) with
given finite-dimensional distribution functions satisfying the consistency conditions.
Theorem 2.4 (Kolmogorov consistency theorem). Suppose a family of distribu-
tion functions F D fFt1 ;:::;tn ; t1 ; : : : ; tn 2 T ; n D 1; 2; : : :g satisfies the consistency
conditions (a) and (b). Then there exists a probability space . ; A; P/, and on that a
stochastic process fXt ; t 2 T g, whose finite-dimensional distributions are identical
to F .
For our considerations, it usually suffices to provide the finite-dimensional
distribution functions of the stochastic processes, in which case the process is
3. 2.3 Stationary Processes 57
defined in a weak sense and it is irrelevant on which probability space it is given.
In some instances the behavior of the random path is significant (e.g., continuity
in time), which is related to a given probability space . ; A; P / where the process
fXt ; t 2 T g is defined. In this case the process is given in a strict sense.
2.3 Stationary Processes
The class of stochastic processes that show a stationary statistical property in time
plays a significant role in practice. Among these processes the most important ones
are the stationary processes in strict and weak senses. The main notions are given
here for one-dimensional processes, but the notion for high-dimensional processes
can be introduced similarly.
Definition 2.5. A process fXt ; t 2 T g is called stationary in a strict sense if the
joint distribution functions of random variables
.Xt1 ; : : : ; Xtn / and .Xt1 Ct ; : : : ; Xtn Ct /
are identical for all t, positive integer n, and t1 ; : : : ; tn 2 T satisfying the conditions
ti C t 2 T ; i D 1; : : : ; n.
Note that this definition remains valid in the case of vector-valued stochastic
processes. Consider a stochastic process X with finite second moment, that is,
E.Xt2 / < 1, for all t 2 T . Denote the expected value and covariance functions by
X .t/ D E .Xt / ; t 2 T ;
RX .s; t/ D cov.Xs ; Xt /
D E ..Xt X .t//.Xs X .s/// ; s; t 2 T :
Definition 2.6. A process fXt ; t 2 T g is called stationary in a weak sense if Xt
has finite second moment for all t 2 T and the expected value and covariance
function satisfy the following relation:
X .t/ D X; t 2T;
RX .s; t/ D RX .t s/; s; t 2 T :
The function RX is called the covariance function.
It is clear that if a stochastic process with finite second moment is stationary
in a strict sense, then it is stationary in a weak sense also, because the expected
value and covariance function depend also on the two-dimensional joint distribution,
4. 58 2 Introduction to Stochastic Processes
which is time-invariant if the time shifts. Besides the covariance function RX .t/, the
correlation function rX .t/ is also used, which is defined as follows:
1 1
rX .t/ D RX .t/ D 2 RX .t/:
RX .0/ X
2.4 Gaussian Process
In practice, we often encounter stochastic processes whose finite-dimensional
distributions are Gaussian. These stochastic processes are called Gaussian. In
queueing theory Gaussian processes often appear when asymptotic methods are
applied.
Note that the expected values and covariances determine the finite-dimensional
distributions of the Gaussian process; therefore, it is easy to verify that a Gaussian
process is stationary in a strict sense if and only if it is stationary in a weak sense. We
also mention here that the discrete-time Gaussian process consists of independent
Gaussian random variables if these random variables are uncorrelated.
2.5 Stochastic Process with Independent and Stationary
Increments
In several practical modeling problems, stochastic processes have independent and
stationary increments. These processes play a significant role both in theory and
practice. Among such processes the Wiener and the Poisson processes are defined
below.
Definition 2.7. If for any integer n 1 and parameters t0 ; : : : ; tn 2 T ; t0 < : : : <
tn , the increments
X t1 X t0 ; : : : ; X tn X tn 1
of a stochastic process X D fXt ; t 2 T g are independent random variables, then
X is called a stochastic process with independent increments. The process X has
stationary increments if the distribution of Xt Ch Xt ; t; t C h 2 T does not
depend on t.
2.6 Wiener Process
As a special but important case of stochastic processes with independent and
stationary increments, we mention here the Wiener process (also called process
of Brownian motion), which gives the mathematical model of diffusion. A process
5. 2.7 Poisson Process 59
X D fXt ; t 2 Œ0; 1/g is called a Wiener process if the increments of the process
are independent and for any positive integer n and 0 Ä t0 < : : : < tn the joint
density function of random variables Xt0 ; : : : ; Xtn can be given in the form
f .x0 ; : : : ; xn I t0 ; : : : ; tn / D .2 / n=2
Œt0 .t1 t0 / : : : .tn tn 1 / 1=2
 2 Ã
1 x0 .x1 x0 /2 .xn xn 1 /2
exp C C:::C :
2 t0 t1 t0 tn tn 1
It can be seen from this formula that the Wiener process is Gaussian and the
increments
Xtj Xtj 1 ; j D 1; : : : ; n;
are independent Gaussian random variables with expected values 0 and variances
tj tj 1 : The expected value function and the covariance function are determined as
X .t/ D 0; RX .s; t/ D min.t; s/; t; s 0:
2.7 Poisson Process
2.7.1 Definition of Poisson Process
Besides the Wiener process defined above, we discuss in this chapter another
important process with independent increments in probability theory, the Poisson
process. This process plays a fundamental role not only in the field of queueing
theory but in many areas of theoretical and applied sciences, and we will deal with
this process later as a Markov arrival process, birth-and-death process, and renewal
process. Its significance in probability theory and practice is that it can be used
to model different event occurrences in time and space in, for example, queueing
systems, physics, insurance, population biology. There are several introductions
and equivalent definitions of the Poisson process in the literature according to its
different characterizations. First we present the notion in the simple (classical) form
and after that in a more general context.
In queueing theory, a frequently used model for the description of the arrival
process of costumers is as follows. Assume that costumers arrive at the system one
after another at t1 < t2 < : : :; tn ! 1 as n ! 1. The differences in occurrence
times, called interarrival times, are denoted by
X1 D t1 ; X2 D t2 t1 ; : : : ; Xn D tn tn 1 ; : : : :
Define the process fN.t/; t 0g with N.0/ D 0 and
N.t/ D maxfn W tn Ä tg D maxfn W X1 C : : : C Xn Ä tg; t > 0:
6. 60 2 Introduction to Stochastic Processes
This process counts the number of customers arriving at the system in the time
interval .0; t and is called the counting process for the sequence t1 < t2 < : : :.
Obviously, the process takes nonnegative integer values only, is nondecreasing, and
N.t/ N.s/ equals the number of occurrences in the time interval .s; t for all
0 < s < t.
In the special case, when X1 ; X2 ; : : : is a sequence of independent and identically
distributed random variables with exponential distribution Exp. /, the increments
N.t/ N.s/ have a Poisson distribution with the parameter .t s/. In addition,
the counting process N.t/ possesses an essential property, that is, it evolves in time
without aftereffects. This means that the past and current occurrences have no
effect on subsequent occurrences. This feature leads to the property of independent
increments.
Definition 2.8. We say that the process N.t/ is a Poisson process with rate if
1. N.0/ D 0,
2. N.t/; t 0 is a process with independent increments,
3. The distribution of increments is Poisson with the parameter .t s/ for all
0 < s < t.
By definition, the distributions of the increments N.t C h/ N.t/, t 0, h > 0,
do not depend on the moment t; therefore, it is a process with stationary increments
and is called a homogeneous Poisson process at rate . Next, we introduce the
Poisson process in a more general setting, and as a special case we have the
homogeneous case. After that we will deal with the different characterizations of
Poisson processes, which in some cases can serve as a definition of the process. At
the end of this chapter, we will introduce the notion of the high-dimensional Poisson
process (sometimes called a spatial Poisson process) and give its basic properties.
Let fƒ.t/; t 0g be a nonnegative, monotonically nondecreasing, continuous-
from-right real-valued function for which ƒ.0/ D 0.
Definition 2.9. We say that a stochastic process fN.t/; t 0g taking nonnegative
integers is a Poisson process if
1. N.0/ D 0,
2. N.t/ is a process with independent increments,
3. The CDFs of the increments N.t/ N.s/ are Poisson with the parameter ƒ.t/
ƒ.s/ for all 0 Ä s Ä t, that is,
.ƒ.t/ ƒ.s//k
P .N.t/ N.s/ D k/ D e .ƒ.t / ƒ.s//
; k D 0; 1; : : : :
kŠ
Since for any fixed t > 0 the distribution of N.t/ D N.t/ N.0/ is Poisson with
mean ƒ.t/, that is the reason that N.t/ is called a Poisson process. We can state
that the process N.t/ is a monotonically nondecreasing jumping process whose
increments N.t/ N.s/; 0 Ä s < t, take nonnegative integers only and the
increments have Poisson distributions with the parameter .ƒ.t/ ƒ.s//. Thus the
7. 2.7 Poisson Process 61
random variables N.t/; t 0 have Poisson distributions with the parameter ƒ.t/;
therefore, the expected value of N.t/ is E .N.t// D ƒ.t/; t 0, which is called a
mean value function.
We also note that using the property of independent increments, the joint
distribution of the random variables N.t1 /; : : : ; N.tn / can be derived for all positive
integers n and all 0 < t1 < : : : < tn without difficulty because for any integers
0 Ä k1 Ä : : : Ä kn we get
P .N.t1 / D k1 ; : : : ; N.tn / D kn /
D P .N.t1 / D k1 ; N.t2 / N.t1 / D k2 k1 ; : : : ; N.tn / N.tn 1 / D kn kn 1 /
.ƒ.t1 //k1 Y .ƒ.ti /
n
ƒ.ti 1 //ki ki 1
D e ƒ.t1 /
e .ƒ.ti / ƒ.ti 1 //
:
k1 Š i D2
.ki ki 1 /Š
Since the mean value function ƒ.t/ D E .N.t// is monotonically nondecreasing,
the set of discontinuity points f n g of ƒ.t/ is finite or countably infinite. It can
happen that the set of discontinuity points f n g has more than one convergence point,
and in this case we cannot give the points of f n g as an ordered sequence 1 < 2 <
: : :. Define the jumps of the function ƒ.t/ at discontinuity points n as follows:
n D ƒ. n C 0/ ƒ. n 0/ D ƒ. n / ƒ. n 0/:
By definition, the increments of a Poisson process are independent; thus it is easy to
check that the following decomposition exists:
N.t/ D Nr .t/ C Ns .t/;
where Nr .t/ and Ns .t/ are independent Poisson processes with mean value
functions
X X
ƒr .t/ D ƒ.t/ n and ƒs .t/ D n:
n <t n <t
The regular part Nr .t/ of N.t/ has jumps equal to 1 only, whose mean value
function ƒr .t/ is continuous. Thus we can state that the process Nr .t/ is continuous
in probability, that is, for any point t, 0 Ä t < 1, the relation
lim P .Nr .t C s/ Nr .t/ > 0/ D lim P .Nr .t C s/ Nr .t/ 1/ D 0
s!0 s!0
is true. The second part Ns .t/ of N.t/ is called a singular Poisson process because
it can have jumps only in discrete points f n g. Then
k
P .Ns . n / Ns . n 0// D k/ D n
e n
; k D 0; 1; 2; : : : :
kŠ
8. 62 2 Introduction to Stochastic Processes
Definition 2.10. If the mean value function ƒ.t/ of a Poisson process
fN.t/; tR 0g is differentiable with the derivative .s/; s 0 satisfying
t
ƒ.t/ D 0 .s/ds, then the function .s/ is called a rate (or intensity) function of
the process.
In accordance with our first definition (2.8), we say that the Poisson process N.t/
is homogeneous with the rate if the rate function is a constant .t/ D ; t 0.
In this case, ƒ.t/ D t; t 0 is satisfied; consequently, the distributions of all
increments N.t/ N.s/; 0 Ä s < t are Poisson with the parameter .t s/ and
E .N.t/ N.s// D .t s/. This shows that the average number of occurrences
is proportional to the length of the corresponding interval and the constant of
proportionality is . These circumstances justify the name of the rate .
If the rate can vary with time, that is, the rate function does not equal a constant,
the Poisson process is called inhomogeneous.
2.7.2 Construction of Poisson Process
The construction of Poisson processes plays an essential role both from a theoretical
and a practical point of view. In particular, it is essential in simulation methods. The
Poisson process N.t/ and the sequence of the random jumping points t1 < t2 < : : :
of the process uniquely determine each other. This fact provides an opportunity to
give another definition of the Poisson process on the real number line. We prove that
the following two constructions of Poisson processes are valid (see, for example, pp.
117–118 in [85]).
Theorem 2.11 (Construction I). Let X1 ; X2 ; : : : be independent and identically
distributed random variables whose common CDF is exponential with parameter 1.
Define
X1
M.t/ D IfX1 C:::CXm Ät g ; t 0: (2.1)
mD1
Then the process M.t/ is a homogeneous Poisson process with an intensity rate
equal to 1.
Theorem 2.12 (Construction II). Let U1 ; U2 ; : : : be a sequence of independent and
identically distributed random variables having common uniform distribution on the
interval .0; T /, and let N be a random variable independent of Ui with a Poisson
distribution with the parameter T . Define
X
N
N.t/ D IfUm Ät g ; 0 Ä t Ä T: (2.2)
mD1
Then N.t/ is a homogeneous Poisson process on the interval Œ0; T at rate .
9. 2.7 Poisson Process 63
We begin with the proof of Construction II. Then, using this result, we verify
Construction I.
Proof (Construction II). Let K be a positive integer and t1 ; : : : ; tK positive constants
such that t0 D 0 < t1 < t2 < : : : < tK D T . Since, by Eq. (2.2), N.T / D N
PN
and N.t/ D IfUm Ät g ; the increments of N.t/ on the intervals .tk 1 ; tk ; k D
mD1
1; : : : ; K, can be given in the form
X
N
N.tk / N.tk 1 / D Iftk 1 <Un Ätk g
; k D 1; : : : ; K:
nD1
Determine the joint characteristic function of the increments N.tk / N.tk 1 /. Let
sk 2 R; k D 1; : : : ; K, be arbitrary; then
( )!
X
K
'.s1 ; : : : ; sK / D E exp i sk .N.tk / N.tk 1 //
kD1
( )ˇ !
1
X X
K ˇ
ˇ
D P .N D 0/ C E exp i sk .N.tk / N.tk 1 // ˇ N D n P .N D n/
ˇ
nD1 kD1
1
( )!
X X
K X
n
De T
C E exp i sk Iftk 1 <U` Ätk g
P .N D n/
nD1 kD1 `D1
1 n
( )!
XY X
K
. T /n
De T
C E exp i sk Iftk 1 <U` Ätk g
e T
nD1 `D1
nŠ
kD1
1
!n ( )
X X
K
tk tk 1 i sk . T /n X
K
De T
e De T
exp e i sk
.tk tk 1 / ;
nD0
T nŠ
kD1 kD1
P
K
and using the relation T D tK t0 D .tk tk 1 / we get
kD1
Y
K
˚ «
'.s1 ; : : : ; sK / D exp .tk tk 1 /.ei sk 1/ :
kD1
Since the characteristic function '.s1 ; : : : ; sK / derived here is equal to the joint char-
acteristic function of independent random variables having a Poisson distribution
with the parameters .tk tk 1 /; k D 1; : : : ; K, the proof is complete. t
u
For the proof of Construction I we need the following well-known lemma of
probability theory.
10. 64 2 Introduction to Stochastic Processes
Lemma 2.13. Let T be a positive constant and k be a positive integer. Let
U1 ; : : : ; Uk be independent and identically distributed random variables having a
common uniform distribution on the interval .0; T /. Define by U1k Ä : : : Ä Ukk
the ordered random variables U1 ; : : : ; Uk . Then the joint PDF of random variables
U1k ; : : : ; Ukk is
(
kŠ
; if 0 < t1 Ä t2 Ä : : : Ä tk < T;
fU1k ;:::;Ukk .t1 ; : : : ; tk / D Tk
0; otherwise.
Proof. Since U1k Ä : : : Ä Ukk , it is enough to determine the joint PDF of random
variables U1k ; : : : ; Ukk on the set
K D f.t1 ; : : : ; tk / W 0 Ä t1 Ä : : : Ä tk < T g:
Under the assumptions of the lemma, the random variables U1 ; : : : ; Uk are inde-
pendent and uniformly distributed on the interval .0; T /; thus for every permutation
i1 ; : : : ; ik of the numbers 1; 2; : : : ; k (the number of all permutations is equal to kŠ)
P Ui1 Ä : : : Ä Uik ; Ui1 Ä t1 ; : : : ; Uik Ä tk
D P .U1 Ä : : : Ä Uk ; U1 Ä t1 ; : : : ; Uk Ä tk /;
then
FU1k ;:::;Ukk .t1 ; : : : ; tk / D P .U1k Ä t1 ; : : : ; Ukk Ä tk /
D kŠP .U1 Ä : : : Ä Uk ; U1 Ä t1 ; : : : ; Uk Ä tk /
Zt1 Ztk
1
D kŠ ::: Ifu Ä::::Äuk g duk : : : du1
Tk 1
0 0
Zt1 Zt2 Ztk
kŠ
D k ::: duk : : : du1 :
T
0 u1 uk 1
From this we immediately have
kŠ
fU1k ;:::;Ukk .t1 ; : : : ; tk / D ; .t1 ; : : : ; tk / 2 K;
Tk
which completes the proof. t
u
Proof (Construction I). We verify that for any T > 0 the process M.t/; 0 Ä t Ä T
is a homogeneous Poisson process with rate . By Construction II, N.T / D N;
11. 2.7 Poisson Process 65
where the distribution of random variable N is Poisson with the parameter T .
From Eq. (2.2) it follows that the process N.t/; 0 Ä t < T , can be rewritten in
the form
X
N X
N
N.t/ D IfUm Ät g D IfUnN Ät g ;
mD1 nD1
where for every k 1 and under the condition N.T / D k the random variables
U1 ; : : : Uk are independent and uniformly distributed on the interval .0; T / and
U1k Ä U2k Ä : : : Ä Ukk are the ordered random variables U1 ; : : : Uk . Note that
we used these properties only to determine the joint characteristic function of the
increments. Define
Tn D X1 C : : : C Xn ; n D 1; 2; : : : ;
where, by assumption, X1 ; X2 ; : : : are independent and identically distributed
random variables with a common exponential CDF of parameter 1. Then, using
the relation (2.1), for any 0 Ä t Ä T ,
8 M.T /
1
X < P
IfTn Ät g ; if T T1 ;
M.t/ D IfTn Ät g D
: nD1
nD1 0; if T < T1 :
By the previous note it is enough to prove that
(a) The random variable M.T / has a Poisson CDF with the parameter T ;
(b) For every positive integer k and under the condition M.T / D k, the joint CDF
of the random variables T1 ; : : : ; Tn are identical with the CDF of the random
variables U1k ; : : : ; Ukk .
(a) First we prove that for any positive t the CDF of the random variable M.t/ is
Poisson with the parameter . T /. Since the common CDF of independent and
identically distributed random variables Xi is exponential with the parameter
, the random variable Tn has a gamma.n; / distribution whose PDF (see the
description of gamma distribution in Sect. 1.2.2.) is
( n
n 1 x
fTn .x/ D €.n/ x e ; if x > 0;
0; if x Ä 0:
From the exponential distribution of the first arrival we have
P .M.t/ D 0/ D P .X1 > t/ D e t
:
12. 66 2 Introduction to Stochastic Processes
Using the theorem of the total expected value, for every positive integer k we
obtain
P .M.t/ D k/ D P .X1 C : : : C Xk Ä t < X1 C : : : C XkC1 /
D P .Tk Ä t < Tk C XkC1 /
Zt k
D P .Tk Ä t < Tk C XkC1 jTk D z/ zk 1
e z
dz
€.k/
0
Zt k
D P .t z < XkC1 / zk 1
e z
dz
€.k/
0
Zt k
D e .t z/
zk 1
e z
dz
€.k/
0
k Zt
. t/k . t/k
D e t
zk 1 dz D e t
D e t
I
€.k/ €.k/k kŠ
0
thus the random variable M.t/; t 0 has a Poisson distribution with the
parameter t.
(b) Let T be a fixed positive number and let U1 ; : : : Uk be independent random
variables uniformly distributed on the interval .0; 1/. Denote by U1k Ä : : : Ä
Ukk the ordered random variables U1 ; : : : Uk . Now we verify that for every
positive integer k the joint CDF of random variables T1 ; : : : ; Tk under the
condition M.T / D k is identical with the joint CDF of the ordered random
variables U1k ; : : : ; Ukk (see Theorem 2.3 of Ch. 4. in [48]).
For any positive numbers t1 ; : : : ; tk , the joint conditional CDF of random
variables T1 ; : : : ; Tk given M.t/ D k can be written in the form
P .T1 Ä t1 ; : : : ; Tk Ä tk ; M.T / D k/
P .T1 Ä t1 ; : : : ; Tk Ä tk jM.T / D k/ D :
P .M.T / D k/
By the result proved in part (a), the denominator has the form
. T /k
P .M.T / D k/ D e T
; k D 0; 1; : : : ;
kŠ
while the numerator can be written as follows:
13. 2.7 Poisson Process 67
P .T1 Ä t1 ; : : : ; Tk Ä tk ; M.T / D k/
D P .X1 Ä t1 ; X1 C X2 Ä t2 ; : : : ; X1 C : : : C Xk Ä tk ; X1 C : : : C XkC1 > T /
Zt1 t2
Z u1 t3 .u1 Cu2 /
Z tk .u1 C:::Cuk
Z 1/ Z1
Y
kC1 Á
D ::: e ui
dukC1 : : : du1
0 0 0 0 T .u1 C:::Cuk / i D1
Zt1 t2
Z u1 t3 .u1 Cu2 /
Z tk .u1 C:::Cuk
Z 1/
k .u1 C:::Cuk / .T u1 C:::Cuk /
D ::: e e dukC1 : : : du1
0 0 0 0
Zt1 t2 u1 t3 .u1 Cu2 /
Z Z tk .u1 C:::Cuk
Z 1/
k T
D e ::: duk : : : du1 :
0 0 0 0
Setting v1 D u1 ; v2 D u1 C u2 ; : : : ; vk D u1 C : : : C uk , the last integral takes
the form
Zt1 Zt2 Zt3 Ztk
. T /k T kŠ
e ::: dvk : : : dv1 ;
kŠ Tk
0 v1 v2 vk 1
thus
Zt1 Zt2 Zt3 Ztk
kŠ
P .T1 Ä t1 ; : : : ; Tk Ä tk jM.T / D k/ D k ::: dvk : : : dv1 :
T
0 v1 v2 vk 1
From this we get that the joint conditional PDF of random variables T1 ; : : : ; Tk
given M.T / D k equals the constant value T k , which, by the preceding lemma,
kŠ
is identical with the joint PDF of random variables U1k ; : : : ; Ukk . Using the proof
of Construction II, we obtain that Construction I has the result of a homogeneous
Poisson process at rate on the interval .0; T , and at the same time on the whole
interval .0; 1/, because T was chosen arbitrarily. t
u
2.7.3 Basic Properties of a Homogeneous Poisson Process
Let N.t/; t 0 be a homogeneous Poisson process with a rate . We enumerate
below the main properties of N.t/.
(a) For any t 0 the CDF of N.t/ is Poisson with the parameter t, that is,
. t/k
P .N.t/ D k/ D e t
; k D 0; 1; : : : :
kŠ
14. 68 2 Introduction to Stochastic Processes
(b) The increments of N.t/ N.s/; 0 Ä s < t, are independent and have a Poisson
distribution with the parameter .t s/.
(c) The sum of two independent homogeneous Poisson processes N1 .tI 1 / and
N2 .tI 2 / at rates 1 and 2 , respectively, is a homogeneous Poisson process
with a rate . 1 C 2 /.
(d) Given 0 < t < T < 1, a positive integer N0 and an integer k satisfy the
inequality 0 Ä k Ä N0 . The conditional CDF of the random variable N.t/
given N.T / D N0 is binomial with the parameters .N0 ; 1=T /.
Proof.
P .N.t / D k; N.T / D N0 /
P .N.t / D k j N.T / D N0 / D
P .N.T / D N0 /
P .N.t / D k; N.T / N.t / D N0 k/
D
P .N.T / D N0 /
 à 1
. t /k . .T t //N0 k . T /N0
D e t e .T t/
e T
kŠ .N0 k/Š N0 Š
! à  Ã
N0 t k t N0 k
D 1 :
k T T
t
u
(e) The following asymptotic relations are valid as h ! C0:
P .N.h/ D 0/ D 1 h C o.h/;
P .N.h/ D 1/ D h C o.h/;
P .N.h/ 2/ D o.h/: .orderliness/
Lemma 2.14. For every nonnegative integer m the inequality
ˇ ˇ
ˇ X xk ˇ
m
jxjmC1 jxj
ˇ x ˇ
ˇe ˇ< e D o.jxjm /; x ! 0;
ˇ kŠ ˇ .m C 1/Š
kD0
holds.
Proof. The assertion of the lemma follows from the nth-order Taylor approximation
to ex with the Lagrange form of the remainder term (see Sect. 7.7 of [4]), but one
can obtain it by simple computations. Using the Taylor expansion
1
X xk
ex D
kŠ
kD0
15. 2.7 Poisson Process 69
of the function ex , which implies that
ˇ ˇ ˇ 1
ˇ
ˇ
ˇ
ˇ X xk ˇ ˇ X xk ˇ
m mC1 X1
.m C 1/Š
ˇ x
ˇe
ˇ ˇ
ˇ Dˇ ˇ Ä jxj jxjk <
ˇ kŠ ˇ ˇ kŠ ˇ .m C 1/Š
ˇ .m C 1 C k/Š
kD0 kDmC1 kD0
1
jxjmC1 X jxjk jxjmC1 jxj
< D e D o.jxjm /; x ! 0:
.m C 1/Š kŠ .m C 1/Š
kD0
t
u
Proof of Property (e). From the preceding lemma we have as h ! C0
P .N.h/ D 0/ D e h
D1 h C o.h/;
h
P .N.h/ D 1/ D e h D h.1 h C o.h// D h C o.h/;
1Š
 à  Ã
. h/1 . h/1
P .N.h/ 2/ D 1 e hC e h D e h 1C e h
D o.h/:
1Š 1Š
t
u
(f) Given that exactly one event of a homogeneous Poisson process [N.t/; t 0]
has occurred during the interval .0; t, the time of occurrence of this event is
uniformly distributed over .0; t.
Proof of Property (f). Denote by the rate of the process N.t/: Immediate
application of the conditional probability gives for all 0 < x < t
P .X1 Ä x; N.t/ D 1/
P .X1 Ä xjN.t/ D 1/ D
P .N.t/ D 1/
P .N.x/ D 1; N.t/ N.x/ D 0/
D
P .N.t/ D 1/
P .N.x/ D 1/P .N.t x/ D 0/
D
P .N.t/ D 1/
! à 1
. x/1 xŒ .t x/0 . t/1 x
D e e .t x/
e t
D :
1Š 0Š 1Š t
t
u
(g) Strong Markov property. Let fN.t/; t 0g be a homogeneous Poisson
process with the rate , and assume that N.t/ is At measurable for all t 0,
where At A; t 0, is a monotonically increasing family of -algebras.
Let be a random variable such that the condition f Ä tg 2 At holds for all
t 0. This type of random variable is called a Markov point with respect to
16. 70 2 Introduction to Stochastic Processes
the family of -algebra At ; t 0. For example, the constant D t and the so-
called first hitting time, k D sup fs W N.s/ < kg, where k is a positive integer,
are Markov points. Denote
N .t/ D N.t C / N. /; t 0:
Then the process N .t/; t 0, is a homogeneous Poisson process with
the rate , which does not depend on the Markov point or on the process
fN.t/; 0 Ä t Ä g.
(h) Random deletion (filtering) of a Poisson process. Let N.t/; t 0 be a
homogeneous Poisson process with intensity > 0. Let us suppose that
we delete points in the process N.t/ independently with probability .1 p/,
where 0 < p < 1 is a fixed number. Then the new process M.t/; t 0,
determined by the undeleted points of N.t/ constitutes a homogeneous Poisson
process with intensity p .
Proof of the Property (h). Let us represent the Poisson process N.t/ in the form
1
X
N.t/ D Iftk Ät g ; t 0;
kD1
where tk D X1 C : : : C Xk ; k D 1; 2; : : : and X1 ; X2 ; : : : are independent
exponentially distributed random variables with the parameter . The random
deletion in the process N.t/ can be realized with the help of a sequence of
independent and identically distributed random variables I1 ; I2 ; : : : ; which do not
depend on the process N.t/; t 0 and have a distribution P .Ik D 1/ D p,
P .Ik D 0/ D 1 p. The deletion of a point tk in the process N.t/ happens only in
the case Ik D 0. Let T0 D 0, and denote by 0 < T1 < T2 < : : : the sequence of
remaining points. Thus the new process can be given in the form
1
X 1
X
M.t/ D IfTk Ät g D Iftk Ät; Ik D1g ; t 0:
kD1 kD1
Using the property of the process N.t/ and the random sequence Ik ; k 1, it
is clear that the sequence of random variables Yk D Tk Tk 1 ; k D 1; 2; : : :,
are independent and identically distributed; therefore, it is enough to prove that they
have an exponential distribution with the parameter p , i.e., P .Yk < y/ D 1 ep y .
The sequence of the remaining points Tk can be given in the form Tk D tnk ; k D
1; 2; : : : ; where the random variables nk are defined as follows:
n1 D minfj W j 1; Ij D 1g;
nk D minfj W j > nk 1; Ij D 1g; k 2:
17. 2.7 Poisson Process 71
Let us compute the distribution of the random variable
Y 1 D T1 D X 1 C : : : C X n 1 :
By the use of the formula of total probability, we obtain
P .Y1 < y/ D P .X1 C : : : C Xn1 < y/
1
X
D P .X1 C : : : C Xn1 < yjn1 D k/P .n1 D k/
kD1
1
X
D P .X1 C : : : C Xk < y/P .n1 D k/:
kD1
The sum X1 C : : : C Xk of independent exponentially distributed random variables
Xi has a gamma distribution with the density function
k
f .yI k; / D yk 1
e y
; y > 0;
.k 1/Š
whereas, on the other hand, the random variable n1 has a geometric distribution with
the parameter p, i.e.,
P .n1 D k/ D .1 p/k 1
pI
therefore, we get
XZ
y
1
X 1 k
P .X1 C : : : C Xk < y/P .n1 D k/ D xk 1
e x
.1 p/k 1 pdx
.k 1/Š
kD1 kD1 0
Zy X
1
! Zy
Œ.1 p/ xk
D p e x
dx D p e.1 p/ x
e x
dx
kŠ
0 kD0 0
Zy
D p e p x
dx D 1 e p y
:
0
t
u
(1) Modeling an inhomogeneous Poisson process. Let fƒ.t/; t 0g be a non-
negative, monotonically nondecreasing, continuous-from-left function such that
ƒ.0/ D 0. Let N.t/; t 0, be a homogeneous Poisson process with rate 1.
Then the process defined by the equation
Nƒ .t/ D N.ƒ.t//; t 0;
18. 72 2 Introduction to Stochastic Processes
is a Poisson process with mean value function ƒ.t/; t 0.
Proof of Property (i). Obviously, N.ƒ.0// D N.0/ D 0, and the increments of
the process Nƒ .t/ are independent and the CDF of the increments are Poissonian,
because for any 0 Ä s Ä t the CDF of the increment Nƒ .t/ Nƒ .s/ is Poisson with
the parameter ƒ.t/ ƒ.s/,
P .Nƒ .t/ Nƒ .s/ D k/ D P .N.ƒ.t// N.ƒ.s///
.ƒ.t/ ƒ.s//k
D e .ƒ.t / ƒ.s//
; k D 0; 1; : : : :
kŠ
t
u
2.7.4 Higher-Dimensional Poisson Process
The Poisson process can be defined, in higher dimensions, as a model of random
points in space. To do this, we first concentrate on the process on the real number
line, from the aspect of a possible generalization.
Let fN.t/; t 0g be a Poisson process on a probability space . ; A; P /.
Assume that it has a rate function .t/; t 0; thus, the mean value function has the
form
Zt
ƒ.t/ D .s/ds; t 0;
0
where the function .t/ is nonnegative and locally integrable function. Denote by
t1 ; t2 ; : : : the sequence of the random jumping points of N.t/. Since the mean value
function is continuous, the jumps of N.t/ are exactly 1; moreover, the process N.t/
and the random points … D ft1 ; t2 ; : : :g determine uniquely each other. If we can
characterize the countable set … of random points ft1 ; t2 ; : : :g, then at the same time
we can give a new definition of the Poisson process N.t/.
Denote by BC D B.RC / the Borel -algebra of the half line RC D Œ0; 1/,
i.e., the minimal -algebra that consists of all open intervals of RC . Let Bi D
.ai ; bi ; i D 1; : : : ; n, be nonoverlapping intervals of RC ; then obviously Bi 2 BC .
Introduce the random variables
˚ «
….Bi / D # f… Bi g D # tj W tj 2 Bi ; i D 1; : : : ; n;
where # f g means the number of elements of a set; then
….Bi / D N.bi / N.ai /:
19. 2.7 Poisson Process 73
By the use of the properties of Poisson processes, the following statements hold:
(1) The random variables ….Bi / are independent because the increments of the
process N.t/ are independent.
(2) The CDF of ….Bi / is Poisson with the parameter ƒ.Bi /, i.e.,
.ƒ.Bi //k
P .….Bi / D k/ D e ƒ.Bi /
;
kŠ
R
where ƒ.Bi / D Bi .s/ds; 1 Ä i Ä n.
Observe that by the definition of random variables ….Bi /, it is unimportant
whether or not the set of random points … D fti g is ordered and ….Bi / is
determined by the number of points ti only, which is included in the interval .ai ; bi .
This circumstance is important because we want to define the Poisson processes on
higher-dimensional spaces, which do not constitute an ordered set, contrary to the
one-dimensional case.
More generally, let Bi 2 B.RC /; 1 Ä i Ä n, be disjoint Borel sets and denote
….Bi / D # f… Bi g. It can be checked that ….Bi / are random variables defined by
the random points … D ft1 ; t2 ; : : :g and they satisfy properties (1) and (2). On this
basis, the Poisson process can be defined in higher-dimensional Euclidean spaces
and, in general, in metric spaces also (see Chap. 2. of [54]).
Consider the d -dimensional Euclidean space S D Rd and denote by B.S /
the Borel -algebra of the subset of S . We will define the Poisson process … as
a random set function satisfying properties (1) and (2). Let … W ! S be a
random point set in S, where S denotes the set of all subsets of S consisting of
countable points. Then the quantities ….A/ D # f… Ag define random variables
for all A 2 B.S /.
Definition 2.15. We say that … is a Poisson process on the space S if … 2 S is a
random countable set of points in S and the following conditions are satisfied:
(1) The random variables ….Ai / D # f… Ai g are independent for all disjoint sets
A1 ; : : : ; An 2 B.S /.
(2) For any A 2 B.S / the CDF of random variables ….A/ are Poisson with the
parameter ƒ.A/, where 0 Ä ƒ.A/ Ä 1.
The function ƒ.A/; A 2 B.S / is called a mean measure of a Poisson process
(see [54], p. 14).
Properties:
1. Since the random variable ….A/ has a Poisson distribution with the parameter
ƒ.A/, then E .….A// D ƒ.A/ and D2 .….A// D ƒ.A/.
2. If ƒ.A/ is finite, then the random variable ….A/ is finite with probability 1, and
if ƒ.A/ D 1, then the number of elements of the random point set … A is
countably infinite with probability 1.
3. For any disjoint sets A1 ; A2 ; : : : 2 B.S /,
20. 74 2 Introduction to Stochastic Processes
1
X 1
X
….A/ D ….Ai / and ƒ.A/ D ƒ.Ai /;
i D1 i D1
where A D [1 Ai . The last relation means that the mean measure ƒ.B/; B 2
i D1
B.S / satisfies the conditions of a measure, i.e., it is a nonnegative, -additive set
function on the measurable space .S; B.S //, which justifies the name of ƒ.
Like the one-dimensional case, when the Poisson process has a rate function, it is
an important class of Poisson processes for which there exists a nonnegative locally
integrable function with the property
Z
ƒ.B/ D .s/ds; B 2 B.S /
B
(here the integral is defined with respect to the Lebesgue measure ds). Then the
mean measure ƒ is nonatomic, that is, there is no point s0 2 B.S / such that
ƒ.fs0 g/ > 0.
4. By the use of properties 1 and 3, it is easy to obtain the relation
1
X 1
X
D2 .….A// D D2 .….Ai // D ƒ.Ai / D ƒ.A/:
i D1 i D1
5. For any B; C 2 B.S /,
cov.….B/; ….C // D ƒ.B C /:
Proof. Since ….B/ D ….B C / C ….BnC / and ….C / D ….B C / C ….C nB/,
where the sets A C; AnC and C nA are disjoint, the ….A C /, ….AnC /, and
….C nA/ are independent random variables, and thus
cov.….A/; ….C // D cov.….A C /; ….A C //
D D2 .….A C // D ƒ.A C /:
t
u
6. For any (not necessarily disjoint) sets A1 ; : : : ; An 2 B.S / the joint distribution
of random variables ….A1 /; : : : ; ….An / is uniquely determined by the mean
measure ƒ.
Proof. Denote the set of the 2n pairwise disjoint sets by
N
C D fC D B1 : : : Bn ; where Bi means the set either Ai , or Ai gI
then the random variables ….C / are independent and have a Poisson distribution
with the parameter ƒ.C /. Consequently, the random variables ….A1 /; : : : ; ….An /
21. 2.7 Poisson Process 75
can be given as a sum from a 2n number of independent random variables
….C /; C 2 C, having a Poisson distribution with the parameter ƒ.C /; there-
fore, the joint distribution of random variables ….Ai / is uniquely determined by
….C /; C 2 C, and the mean measure ƒ. t
u
Comment 2.16. Let S D Rd , and assume
Z
ƒ.A/ D .x/dx; A 2 B.S /;
A
where .x/ is a nonnegative and locally integrable function and dx D dx1::: dxn . If
jAj denotes the n-dimensional (Lebesgue) measure of a set A and the function .x/
is continuous at a point x0 2 S , then
ƒ.A/ .x0 / jAj
if the set A is included in a small neighborhood of the point x0 .
The Poisson process … is called homogeneous if .x/ D for a positive
constant . In this case for any A 2 B.S / the inequality ƒ.A/ D jAj holds.
The following three theorems state general assertions on the Poisson processes
defined in higher-dimensional spaces (see Chap. 2 of [54]).
Theorem 2.17 (Existence theorem). If the mean measure ƒ is nonatomic on the
space S and it is -finite, i.e., it can be expressed in the form
1
X
ƒD ƒi ; where ƒi .S / < 1;
i D1
then there exists a Poisson process … on the space S and has mean measure ƒ.
Theorem 2.18 (Superposition theorem). If …i ; i D 1; 2; : : :, is a sequence of
independent Poisson processes with mean measure ƒ1 ; ƒ2 ; : : : on the space S ,
then the superposition … D [1 …i is a Poisson process with mean measure
P i D1
ƒ D 1 ƒi :
i D1
Theorem 2.19 (Restriction theorem). Let … be a Poisson process on the space S
with mean measure ƒ. Then for any S0 2 B.S / the process
…0 D … S0
can be defined as a Poisson process on S with mean measure
ƒ0 .A/ D ƒ.A S0 /:
The process …0 can be interpreted as a Poisson process on the space S0 with mean
measure ƒ0 , where ƒ0 is called the restriction of mean measure ƒ to S0 .
22. 76 2 Introduction to Stochastic Processes
2.8 Exercises
Exercise 2.1. Let X1 ; X2 ; : : : be independent identically distributed random vari-
ables with finite absolute moment E .jX1 j/ < 1. Let N be a random variable taking
positive integer numbers and independent of the random variable .Xi ; i D 1; 2; : : :/.
Prove that
(a) E .X1 C : : : C XN / D E .X1 / E .N /,
(b) D2 .X1 C : : : C XN / D D2 .X1 / C .E .X1 //2 .E .N //2
(Wald identities or Wald lemma).
Exercise 2.2. Let X0 ; X1 ; : : : be independent random variables with joint distribu-
tion P .Xi D 1/ D P .Xi D 1/ D 1 . 2
Define Z0 D 0, Zk D Zk 1 C Xk ; k D 0; 1; : : :. Determine the expectation and
covariance function of the process .Zk ; k D 1; 2; : : :/ (random walk on the integer
numbers).
Let a and b be real numbers, jbj < 1. Denote W0 D aX0 ; Wk D bWk 1 C
Xk ; k D 1; 2; : : : [here the process .Wk ; k D 0; 1; : : :/ constitutes a first-degree
autoregressive process with the initial value aX0 and with the innovation process
.Xk ; k D 1; 2; : : :/]. If we fix the value b, for which value of a will the process Wk
be stationary in a weak sense?
Exercise 2.3. Let a and b be real numbers, and let U be a random variable
uniformly distributed on the interval .0; 2 /. Denote Xt D a cos.bt C U /, 1 <
t < 1. Prove that the random cosine process .Xt ; 1 < t < 1/ is stationary.
Exercise 2.4. Let N.t/; T 0 be a homogeneous Poisson process with inten-
sity .
(a) Determine the covariance and correlation functions of N.t/.
(b) Determine the conditional expectation E .N.t C s/ j N.t//.