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Consider the following gambler's ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the nth play of the game. Then Xn+1 = = [X₂+1 (Xn - 1 Xn+1 = Xn with probability p with probability 1-p for 0 < X

Consider the following gambler's ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune of T dollars. Let Xn denote the number of dollars possessed by the gambler after the nth play of the game. Then Xn+1 = = [X₂+1 (Xn - 1 Xn+1 = Xn with probability p with probability 1-p for 0 < X

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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4. Consider the following gambler's ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune T dollars. Let Xn denote the number of dollars possessed by the gambler after the nth play of the game. Then Xn+1: SX n + 1 (x₂-1 LXn - for 0 < X <T, or T. {Xn} is a Markov chain. The gambler starts with Xo dollars, where Xo is a positive integer less than T. (a) How many states are here? Construct the (one-step) transition matrix of the Markov chain. What is the dimension of this matrix? (b) Now if T = 4 and p = 0.2, write down the transition matrix. with probability p with probability 1-p Xn+1 = Xn Xn 0, = 0, for X₂
Transcribed Image Text:4. Consider the following gambler's ruin problem. A gambler bets $1 on each play of a game. Each time, he has a probability p of winning and probability 1 - p of losing the dollar bet. He will continue to play until he goes broke or nets a fortune T dollars. Let Xn denote the number of dollars possessed by the gambler after the nth play of the game. Then Xn+1: SX n + 1 (x₂-1 LXn - for 0 < X <T, or T. {Xn} is a Markov chain. The gambler starts with Xo dollars, where Xo is a positive integer less than T. (a) How many states are here? Construct the (one-step) transition matrix of the Markov chain. What is the dimension of this matrix? (b) Now if T = 4 and p = 0.2, write down the transition matrix. with probability p with probability 1-p Xn+1 = Xn Xn 0, = 0, for X₂
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