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STATS320 – Assignment 2 – Due at 6 pm, Monday 3 April, 2023

This assignment contributes 7.5% of your final grade.

Maximum possible marks: 30 marks

1 A family receives the paper every morning and places it on a pile after reading it. Each afternoon, with probability 1/3, someone takes all the papers in the pile and puts them in the recycling bin. Also, if ever there are at least five papers in the pile, someone, with probability 1, takes the pile of papers to the bin. Consider the number of papers in the pile in the evening. Is it reasonable to model this by a Markov chain? If so, what is the state space and the 1-step transition probability matrix?             [6 marks]

2 Suppose {Xn ,n ≥ 0} is a birth and death chain with state space S = {0, 1, 2, 3, 4} and transition probabilities given by

P =         0              µ          1 − µ − 2λ        2λ            0      

(      )

where λ,µ > 0, and 3λ + µ < 1.

(a) Write down equations that the equilibrium distribution satisfies.

(b) Show that the equilibrium distribution for this chain is given by

π0  = ,   πk  =  (   ) k , 1 ≤ k ≤ 4,                   where n Pk  =  and G = 1 +习Pk  (  ) k  is the normalizing constant.

(c) Is this also the limiting distribution? Justify your answer briefly.                         [9 marks]

3 Google’s PageRank algorithm relies on a fairly simple stochastic model of web browsing. According to this model, a given person surfs the web by first choosing a page at random and then jumping to another page in two different ways:

• With probability α, select one of the link available on the current page, with each link having the same probability of being selected.

• With probability 1 − α, select a new page uniformly at random.

The PageRank score of a given page corresponds to the long-term frequency with which this page is visited while surfing the web according to the process just described.

Write a computer program that calculates the PageRank score for the following graph where α = 0.85.

Hint:  You may need to use function order() in R to find the increasing order of most frequently visited pages.                                                                                                                    [15 marks]