A First Course in Probability
9th Edition
ISBN: 9780321794772
Author: Sheldon Ross
Publisher: PEARSON
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Textbook Question
Chapter 9, Problem 9.4PTE
Suppose that 3 white and 3 black balls are distributed in two urns in such a way that each urn contains 3 balls. We say that the system is in state i if the first urn contains i white balls.
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Three white and three black balls are distributed in two urns in such a way that each urn
contains three balls. We say the system is in state i (i = 0, 1, 2, 3) if the first urn contains i white
balls. At each step, we draw one ball firom each urn and place the ball drawn from the first urn
into the second, and conversely with the ball from the second urn. Let X, denote the state of the
system after the nth step. Specify the transition probability matrix for the Markov chain {Xn}.
Four white and four black balls are distributed in two urns in such a way that each contains four balls. We say that the system is in state i,i = 0,1,2,3,4 , if the first urn contains i white balls. At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second, and conversely with the ball from the second urn. Let Xn denote the state of the system after the nth step. Explain why {Xn, n = 1, 2, 3, . . .} is a Markov chain and calculate its transition matrix.
Q3) Three white and three black balls are distributed in two urns in such a way that each urn contains
three balls. We say that the system is in state i, i = 0,1,2,3, if the first urn contains i white balls initially.
At each step, we draw one ball from each urn and place the ball drawn from the first urn into the second,
and conversely with the ball from the second urn. The system is in state j,j = 0,1,2,3, if the first urn
contains j white balls after the exchange. Let Xn denote the state of the system after the nth step.
a) Explain why {X, n = 0,1,2,...} is a Markov chain.
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b) Calculate its transition probability matrix P.
Chapter 9 Solutions
A First Course in Probability
Ch. 9 - Customers arrive at a bank at a Poisson rate ....Ch. 9 - Cars cross a certain point in the highway in...Ch. 9 - Suppose that in Problem 9.2, AI is agile enough to...Ch. 9 - Suppose that 3 white and 3 black balls are...Ch. 9 - Consider Example 2a. If there is a 50-50 chance of...Ch. 9 - Compute the limiting probabilities for the model...Ch. 9 - A transition probability matrix is said to be...Ch. 9 - On any given day, Buffy is either cheerful (c),...Ch. 9 - Suppose that whether it rains tomorrow depends on...Ch. 9 - A certain person goes for a run each morning. When...
Ch. 9 - Prob. 9.11PTECh. 9 - Determine the entropy of the sum that is obtained...Ch. 9 - Prove that if X can take on any of n possible...Ch. 9 - A pair of fair dice is rolled....Ch. 9 - A coin having probability p=23 of coming up heads...Ch. 9 - Prob. 9.16PTECh. 9 - Show that for any discrete random variable X and...Ch. 9 - Prob. 9.18PTECh. 9 - Events occur according to a Poisson process with...Ch. 9 - Prob. 9.2STPECh. 9 - Prob. 9.3STPECh. 9 - Prob. 9.4STPECh. 9 - Prob. 9.5STPE
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- A product is made by only two competing companies. Suppose Company X retains two-thirds of its customers and loses one-third to Company Y each year, and Company Y retains three-quarters of its customers and loses one-quarter to Company X each year. We can represent the number of customers each company had last year by D where xo is the number Company X had and yo is the number Company Y had. Then the number that each will have this year can be represented by 3 If Company X has 2900 customers and Company Y has 2500 customers this year, how many customers did each have last year? Company X Company Y Need Help? Read It customers customersarrow_forwardThe Phony TV company makes two different types of television sets, OLED and LED, which are assembled by two different assembly lines. When Line One operates it assembles 30 units of the OLED model and 50 units of the LED model per hour. Line Two assembles 40 units of the OLED model and 40 units of the LED model per hour. Let x be the number of hours that Line One operates and y the number of hours Line Two operates. Phony needs to produce at least 3000 units of the OLED model and 4000 units of the LED model to fill an order. (a) Write down the inequalities that describe the assembly constraints. (b) Graph the feasible region determined by these constraints.arrow_forward
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