Show that the chain of independent random variables has a unique stationary distribution and findwhat it is.

A probability distribution of the Markov chain which remains stable and unchanged as time progresses…

Answered: Show that the chain of independent…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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CHAPTER 5 TEXTBOOK Section 5.18- Exercise 20 Let X be the number of bacterial colonies per cubic centimeter, a Poisson random variable with expected value 3. (i) What is the probability that there is at least one bacterial colony in a randomly chosen cubic centimeter? (ii) What is the probability that in five randomly chosen cubic centimeters there is at least one cubic centimeter where there is at least one bacterial colony? (iii) How many cubic centimeters must be observed for the probability of observing at least one with at least one bacterial colony to be 0.95? 0.9999997 ÷ What is the probability that there is at least one bacterial colony in a randomly chosen cubic centimeter? What is the probability that in five randomly chosen cubic centimeters there is at least one cubic centimeter where 0.9502 there is at least one bacterial colony? How many cubic centimeters must be observed for the probability of observing at least one with at least one 1 bacterial colony to be 0.95?
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Person P is walking with his dog every day. He decides to model the time intervals between dog encounters (i.e., how long does it take between two encounters) with the independent random variables Y₁, . . ., Yn . Number of dogs encounters per day, the number of customers entering a store per hour, the number of bacteria per liter of water, etc. are classically modelled using the Poisson distribution. With an exponential distribution, on the other hand, is used for modelling intervals of between events and how long a light bulb lasts, for example. After learning this, person P decides to model the observations in such a way that the corresponding random variables Y1,..., Yn for a random sample, that is, independent observation, of the exponential distribution Exp(1/µ), µ > 0. The parameter µ > 0 is used as the statistical model parameter 1 which is the expected value of each observation random variable Yi. (a) Form the maximum likelihood estimate μ^(y) in the model for the parameter μ…
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Show that the chain of independent random variables has a unique stationary distribution and find what it is.