Midterm Exam II
ECE534 Spring 2006
There are a total of five problems April 12, 7:00-8:30 pm
You are allowed one sheet (two pages) of notes; no calculators. Each problem is worth 20 points
Please put your NAME here:
1. Consider an object performing a random walk on the vertices of a square. At each time step,
the object stays at the same vertex that it was at the previous time step with probability 1/3 or
it moves to one of the two adjacent vertices, each with probability 1/3.Let Xbe a discrete-time
random process such that Xndenotes the position of the object at time step n. Name the vertices
of the square A, B, C and D, such that Aand Care diagonally opposite vertices. Assume X0=A.
Let
Tj= min{n≥1 : Xn=j},
where jcan be either A, B, C or D, and let yj=E(Tj).Thus, yjis the expected time to reach
vertex j, starting from vertex A. Note that nis taken to be greater than or equal to 1 in the
definition of Tj.
(a) Is X a Markov process? Explain your answer.
(b) Find yA.(Hint: Let t1be the exp ected time to reach vertex Astarting from either vertex
Bor vertex Dand let t2be the expected time to reach Astarting from C. Find equations relating
yA, t1and t2.)
(c) Find yB, yCand yD.
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