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Markov chains

Markov chains. Probability distributions. Exercise. 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.

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Markov chains

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  1. Markov chains

  2. Probability distributions

  3. Exercise 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. Use the barplot function to plot the pmf values (k=0..10) for Bin(10,0.5) as below:

  4. Markov chains

  5. Markov chains

  6. 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

  7. 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. • Pij 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’)

  8. Weather model example

  9. 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 30th day?

  10. Weather model example

  11. Eigenvectors / Eigenvalues

  12. Eigenvectors / Eigenvalues 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 A’s 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.

  13. Exercise • 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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