MA22S6 Numerical and Data Analysis 1 2015-2016
Homework sheet 7
th
Due Tuesday 12 of April at the latest (hand in your assignment either to me
directly (LLoyd 2.46) or to Helen Murphy in the School of Maths office)
1. Describe what is meant by a Markov process and give the definition of the Markov
matrix.
2. We assume that the weather on a given day in Dublin can be described by a Markov
process with three states: cloudy, sunny or rainy. If today is rainy, then the probability tomorrow is rainy is 70% and the probability tomorrow is cloudy is 20%. If
today is cloudy, then the probability tomorrow is cloudy is 35% and the probability
tomorrow is sunny is 20%. If today is sunny, the probability that tomorrow is sunny
is 50% and that tomorrow is rainy is 15%.
a) Choose the basis (1, 0, 0), (0, 1, 0) and (0, 0, 1) corresponding to the three different states and give the Markov matrix M in this basis.
b) Given that today (t = 0) is cloudy, find the probability tomorrow (t = 1) is
sunny. Same for the day after tomorrow (t = 2) and the day after that (t = 3).
c) On average, what is the probability of having a sunny day? a rainy day?
d) What happens if you choose a different basis?
e) Given the fixed point probability distribution (aka as eigenvector of M to eigenvalue 1) that you have calculated, check whether M satisfies the detailed balance
conditions.
f) Find a different Markov matrix that satisfies the detailed balance conditions
for the given fixed point probability distribution.
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Lecturer: Stefan Sint, sint@maths.tcd.ie