Probability on Graphs Random Processes on Graphs and …

If the two principal characters in these notes are random walk and per-colation, they are only part of the rich theory of uniform spanning trees, self-avoiding walks, random networks, models for ferromagnetism and the spread of disease, and motion in random environments. This is an area that

  Processes, Probability, Graph, Random, Probability on graphs random processes on graphs

TIME SERIES - University of Cambridge

Syllabus Time series analysis refers to problems in which observations are collected at regular time intervals and there are correlationsamong successive observations.

  Series, Time, Time series

Markov Chains - Statistical Laboratory

Markov Chains These notes contain material prepared by colleagues who have also presented this course at Cambridge, especially James Norris. The material mainly comes from books of

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TIME SERIES - University of Cambridge

1 Models for time series 1.1 Time series data A time series is a set of statistics, usually collected at regular intervals. Time series data occur naturally in many application areas.

Lecture 6. Bayesian estimation

In 1763, Reverend Thomas Bayes of Tunbridge Wells wrote In modern language, given r ˘Binomial( ;n), what is P( 1 < < 2jr;n)? Lecture 6. Bayesian estimation 6 (1{72) 6. Bayesian estimation 6.2. Prior and posterior distributions Example 6.1 Suppose we are interested in the true mortality risk in a hospital H which is

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1 Gambler’s Ruin Problem - Columbia University

1 Gambler’s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1−p respectively. Let R n denote the total fortune after the nth gamble. The gambler’s objective is to reach a total

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Chapter 1 Markov Chains - Yale University

Gambler’s Ruin. Consider a Markov chain on S= ... In this case, the outcome of the game depends on the Gambler’s fortune. When the fortune is i, the Gambler either wins or loses $1 with respective probabilities p i or q i, or breaks even (the fortune does not change) with probability r i. Another interpretation is that the state of the ...

  Gamblers, Ruin, Gambler s ruin

ONE-DIMENSIONAL RANDOM WALKS

Gambler’s Ruin. Simple random walk describes (among other things) the fluctuations in a speculator’s wealth when he/she is fully invested in a risky asset whose value jumps by either 1 in each time period. Although this seems far too simple a model to be of any practical value,

  Dimensional, Walk, Random, Gamblers, Ruin, Gambler s ruin, One dimensional random walks

1 Stopping Times - Columbia University

1. (First passage/hitting times/Gambler’s ruin problem:) Suppose that X has a discrete state space and let ibe a xed state. Let ˝= minfn 0 : X n= ig: This is called the rst passage time of the process into state i. Also called the hitting time of the process to state i. More generally we can let Abe a collection of states such

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Chapter 6 - Random Processes - UAH

random walk have been studied over the years (i.e., the gambler's ruin, drunken sailor, etc.). At first, a discrete random walk is introduced. Then, it is shown that a limiting form of the random walk is the well-known continuous Wiener process. Finally, simple equations are developed that

  Gamblers, Ruin, Gambler s ruin

Introduction to Stochastic Calculus - Duke University

i) The gambler’s ruin problem We play the following game: We start with 3$ in our pocket and we ip a coin. If the result is tail we loose one dollar, while if the result is positive we win one dollar. We stop when we have no money to bargain, or when we reach 9$. We may ask: what is the probability that I end up broke?

  Introduction, Calculus, Stochastic, Gamblers, Ruin, Gambler s ruin, Introduction to stochastic calculus

Essentials of Stochastic Processes - Duke University

To check this for the gambler’s ruin chain, we note that if you are still playing at time n, i.e., your fortune X n = i with 0 < i < N, then for any possible history of your wealth i n−1,i n−2,...i 1,i 0 P(X n+1 = i+1|X n = i,X n−1 = i n−1,...X 0 = i 0) = 0.4 since to increase your wealth by one unit you have to win your next bet. Here

  Gamblers, Ruin, Gambler s ruin

Random Walk: A Modern Introduction - University of Chicago

4.6 Green’s function for a set 96 5 One-dimensional walks 103 5.1 Gambler’s ruin estimate 103 5.1.1 General case 106 5.2 One-dimensional killed walks 112 5.3 Hitting a half-line 115 6 Potential Theory 119 6.1 Introduction 119 6.2 Dirichlet problem 121 6.3 Difference estimates and Harnack inequality 125 6.4 Further estimates 132

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