1. Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of itsbeing clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that these probabilities do not change ifinformation is also provided about the weather before today.(a) Formulate the evolution of the weather as a Markov chain by defining its states and giving its (one-step)transition matrix.(b) Find the n-step transition matrix P(n) for n = = 2, 5, 10, 20.(c) The probability that it will rain today is 0.4. Use the results from part (b) to determine the probability that itwill rain n days from now, for n = 2, 5, 10, 20.

a)The weather can be modeled as a Markov chain with two states: "rainy" and "clear" (or "not…

Answered: 1. Assume that the probability of rain…

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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Determine whether the Markov chain with matrix of transition probabilities P is absorbing. Explain.
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5
Question 4(a) Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of its being clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that these probabilities do not change if information is also provided about the weather before today.(i) Explain why the stated assumptions imply that the Markovian property holds for the evolution of the weather. (ii) Formulate the evolution of the weather as a Markov chain by defining its states and giving its (one-step) transition matrixb) i. consider the M/M/1 queue with arrival rate λ and service rate µ and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of , the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution d) If an M/D/1 queue has utilization of 80% do you expect its mean queue…
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