STAT 333 Assignment 3
Due: Monday April 2 at the beginning of class
1.
At all times, a container holds a mixture of N balls, some white and the rest black. At each
step, a coin having probability p, 0 < p < 1, of landing heads is tossed. If it is heads, a ball is
chosen at random from the container and replaced by a white ball; if it is tails, a ball is
chosen at random from the container and replaced by a black ball. Let X
n
denote the number
of white balls in the container after the n
th
coin toss.
a.
Is {X
n
} 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 p
ij
.
d.
Let N = 2. Find the limiting proportion of time that {X
n
} spends in each state.
e.
Based on your answer in part (d) and your intuition, guess the answer for the limiting
probability when N is any positive integer.
f.
Prove your guess in part (e) by showing that the equilibrium equations are satisfied.
g.
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 Winter '08
 Chisholm
 Probability, Probability theory, #, 4 minutes, 3 minutes, 4 min, exponential rate

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