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Markov chains

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Probability distributions

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Exercise 1.Use the Matlab function nchoosek(n,k) to implement a generic function BinomialPMF(k,n,p) for calculating the Binomial PMF with k successes in n trials with probability p. 2.Use the barplot function to plot the pmf values (k=0..10) for Bin(10,0.5) as below:

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Markov chains

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Example: A very simple weather model (modified from Wikipedia page Examples of Markov chains) The probabilities of weather conditions, modeled as either sunny=0 or rainy=1, given the weather on the preceding day, can be represented by a transition matrix

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Weather model example The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. The columns can be labelled sunny and rainy respectively, and the rows can be labelled in the same order. P ij is the probability that, if a given day is of type i, it will be followed by a day of type j. Note that the rows of P add up to 1: sum(P)

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Weather model example

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Exercise If the the weather on the first day has 50% probability of being sunny or rainy, then what is the probability of the ninth day being sunny? What about the 30 th day?

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Weather model example

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Eigenvectors / Eigenvalues

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We can calculate eigenvalues and eigenvectors in Matlab using the built-in function eig. The default behavior is for right eigenvalues/eigenvectors, but left eigenvalues/eigenvectors are easily obtained by transposing the matrix. We will be using the form (from Matlab help for eig): [V,D] = eig(A) produces matrices of eigenvalues (D) and eigenvectors (V) of matrix A, so that A*V = V*D. Matrix D is the canonical form of A a diagonal matrix with As eigenvalues on the main diagonal. Matrix V is the modal matrix - its columns are the eigenvectors of A. Note that Matlab always returns eigenvectors with norm 1.

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Exercise 1.Use the eig function to find out, in the long term, what is the percentage of sunny days in the weather example.

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