Stat 430
Homework 8
Fall 2011
Maximum score is 30 points, due date is Tuesday, Nov 15th. You can either hand in the solution electronically
or on paper during class.
1
Weather in Oz
According to Kemeny, Snell, and Thompson, the Land of Oz is blessed by many things, but not by good
weather:? They never have two nice days in a row. ? If they have a nice day, they are just as likely to have
snow as rain the next day. ? If they have snow or rain, they have an even chance of having the same the
next day. ? If there is change from snow or rain, only half of the time is this a change to a nice day. ??
(a) Take as states the kinds of weather R, N, and S and derive transition probabilities from the information
above.
(b) Given that on day 3 it is snowing, what is the probability that it will be nice on day 5? What is the
probability that it is nice on day 6 when it is snowing on day 3?
(c) In R, set up the transition matrix P. Write a function MatrixPowers with argument k that returns the
k step transition matrix. (Hint: matrix multiplication in R is done with %*%)
Compute and compare the results from MatrixPowers for k = 1, ..., 10
(d) Using MatrixPowers, compute the probability that it is nice on day 15, if it is nice on day 3.
(10 points)
2
Drunkard’s Walk
cf. Grinstead & Snell’s, Introduction to Probability
There are 4 blocks between a man’s home and his favorite bar. At each corner the man chooses to go in the
direction of his home or to the bar with equal probability. If he reaches either home or the bar, he stays
there.
Denote each of the corner by a state, with 0 = home and 4 = bar.
(a) Set up a transition probability matrix P from the information above.
(b) If the man starts at corner 1, what is the probability that he is at home at step 3 ? Use MatrixPowers
(see previous question) to compute probabilities for the first 5 steps. What do you observe? Describe.
(c) If the man starts in corner 1, what is the probability that he will reach the bar (at some time)?
(d) Compute the fundamental matrix of P , and give the number of expected steps until the man reaches
either bar or home.
(10 points)
3
Loaded Coin
adapted from Grinstead & Snell’s, Introduction to Probability
Consider the following process. We have two coins, one of which is fair, and the other is ‘loaded’, i.e. it
prefers to show heads at a rate of 80 to 20. We give these two coins to our friend, who chooses one of them
at random (each with probability 1/2). During the rest of the process, she uses only the coin that she chose.
She now proceeds to toss the coin many times, reporting the results. We consider this process to consist
solely of what she reports to us.
(a) Given that she reports a head on the nth toss, what is the probability that a head is thrown on the
(n + 1)st toss?
(b) Consider this process as having two states, heads and tails. By computing the other three transition
probabilities analogous to the one in part (a), write down a “transition matrix” for this process.
(c) Now assume that the process is in state “heads on both the (n − 1)st and the nth toss. Find the
probability that a head comes up on the (n + 1)st toss.
(d) Is this process a Markov chain?
(10 points)
2