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356 Class Note for STAT 416 at PSU

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15
Exercises for Chapter 4 Markov Chain 1 A particle moves on a circle through points which have been marked 0 l 2 3 4 in a clockwise or der At each step it has a probability p of moving to the right and l p to the left Let X denote its location on the circle after the nth step The process Xm n 2 0 is a Markov chain a Find the transition probability matrix b Is this MC irreducible Find all the recurrent states and transient states 2 Each of 2 switches is either on or off during a day On day n each switch will independently be on wit probability 1 on sw1tchis in day n 1 What frac tion of days are both switches on both off 3 Three out of every 4 trucks on the road are followed by a car while only 1 out of every 5 cars is followed by a truck What fraction of vehicles on the road are trucks 4 Each morning an individual leaves his house and goes for a run He is equally likely to leave either from his front or back door Upon leaving the house he 1 chooses a pair of running shoes or goes running bare foot if there are no shoes at the door from which he departed On his return he is equally likely to en ter and leave his running shoes either by the front or back door If he owns a total of 19 pairs of running shoes what proportion of the time does he run bare footed Suppose Max and Patty decide to ip pennies The one coming closest to the wall wins Patty being the better player has a probability of 06 of winnig on each ip If Patty starts with 5 pennies and Max with 10 then what is the probability that Patty will wipe Max out A total of m white and m black balls are distributed among two urns with each urn containing m balls At each stage a ball is randomly selected from each urn and the two selected balls are interchanged Let X n denote the number of black balls in urn 1 after the nth interchange a Give the transition probabilities of the Markov chain X n 2 0 b Without any computations what do you think are the limiting probabilities of this chain c Find the limiting probabilities and show that the stationary chain is time reversible 2 7 At all times an urn contains N balls some white balls and some black balls At each stage a coin hav ing probability p 0 lt p lt l of landing heads is ipped If heads appears then a ball is chosen at ran dom from the urn and is replaced by a white ball if tails appears then a ball is chosen from the urn and is replaced by a black ball Let X denote the number of white balls in the urn after the nth stage a Is an 2 0 a Markov chain If so explain why b What are its classes What are their periods Are they transient or recurrent c Compute the transition probabilities Pm d Let N 2 Find the proportion of time in each state e Based on your answer in part d and your intu ition guess the answer for the limiting probability in the general case i Prove your guess in part e g If p 1 what is the expected time until there are only white balls in the urn if initially there are i 1 white and N z39 4 l 3 black What is the expected time for a general N and 2

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