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Quiz

MATH 303 Quiz: HW2Exam


Department
Mathematics
Course Code
MATH 303
Professor
Marco Ferrante
Study Guide
Quiz

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MATH 303 – Assigment 2: Due Wednesday, January 23 at start of class
I. Problems to be handed in:
1. A Markov chain {Xn, n 0}with state space S={0,1,2,3,4,5}has transition proba-
bility matrix
P=
α0 1 α0 0 0
β/2 0 0 1/3 2/3β β/2
0 0 1 0 0 0
0β/2β/2 1/2 0 0
0 0 0 1 γ0γ
γ1γ0 0 0 0
(a) Determine the equivalence classes of communicating states for any possible choice of
the three parameters α, β and γ;
(b) In all cases, determine if the states in each class are recurrent or transient and find
their period (or determine that they are aperiodic).
2. An equivalent class C of communicating states, called a communicating class, is said to
be closed if Pi,j = 0 whenever iCand j6∈ C. In other words, a communicating class
is closed if there is no escape from that class.
(a) Show that every recurrent class is closed.
(b) Show that every Markov chain on a finite state space has at least one closed com-
municating class.
(c) Find an example of a Markov chain with no closed communicating class.
3. Divide the circle into Npoints, and consider a particle randomly moving between these
points, in such a way that the position of the particle is a Markov Chain, and at each
step the particle jumps to a neighboring point with equal probabilities:
1
2
1
2
0
1
2
···
N1
Compute P(Xn= 0|X0= 0) for all n0. Hint: Use the discrete Fourier transform to
diagonalize the transition matrix. That is, express the transition matrix in the orthonor-
mal basis ˆe(0),ˆe(1),...,ˆe(N1) of CNwhich has the following components in the standard
basis: ˆe(k)
j=1
Ne2πi
Njk, (j= 0, . . . , N 1, i=1).
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