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math_380_f_s1_41corr.pdf - Final Exam, S1 1441 M 380 -...

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1ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــAnswer the following questions:Q1: [4+4](a)Consider the Markov chain whose transition probability matrix is given by0123010001 0.10.40.10.42 0.20.10.60.130001P=(i) Starting in state 2, determine the probability that the Markov chain ends in state 0.(ii) Determine the mean time to absorption.b) LetnXdenote the quality of the nth item that produced in a certain factory with0nmeaning “good” and1nXmeaning “defective”. SupposethatnXbe a Markov chain whosetransition matrix is00 0.990.01P1 0.120.88i) What is the probability that the fourth item is defective given that the first item is defective?ii) In the long run, what is the probability that an item produced by this system is good?Q2: [4+4](a) The following experiment is performed: An observation is made of a Poisson random variableNwith parameter.ThenNindependent Bernoulli trials are performed, each with probabilitypof success. LetZbe the total number of successes observed in theNtrials.i) FormulateZas a random sum and thereby determine its mean and variance.ii)What is the distribution ofZX1

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Term
Fall
Professor
NoProfessor
Tags
Probability theory, Markov chain, Andrey Markov, King Saud University, Department of Mathematics

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