Probability

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Excercises: Statistics 18. lokakuuta 2010
2 1. Let y be a (univariate) Gaussian random variable y ∼ N (µy , σy ). Select the expectation µy 2 and variance σy , and:(a) Plot the probability density function π(y). (b) Draw samples from the distribution of y (c) Draw histogram of the samples. Check that the histograms correspond toπ(y).
2 Repeat this selecting different values for expectation µy and variance σy :

Notes: You need Matlab functions: randn and hist. For drawing samples fromGaussian distribution, see Section 4.1. in the book. 2. Let y ∈ R2 , y = (y1 , y2 ) be a two-dimensional Gaussian random variable y ∼ N (µy , Γy ). Select the expectation µyand covariance Γy , and: (a) Plot the probability density function π(y). (b) Draw samples from the distribution of y. (c) Plot all the samples in the same figure withπ(y). Again, repeat this using different expectations µy and covariances Γy : Notes: Try vectors y with both mutually uncorrelated and correlated elements y1 and y2 . Fordrawing samples from multivariate Gaussian distribution, see Equation (4.5) in the book. 3. Conditional densities (...Continues from Excercise 2...): Select theexpectation µy and covariance Γy , and: (a) plot the conditional density π(y2 |y1 ), corresponding to a few different values of y1 . (b) Draw samples from the conditionaldistribution of y, corresponding to selected values of y1 . (c) In each case (different values of y1 ), draw histogram of the samples. Check that the histograms correspond toπ(y2 |y1 ). Again, repeat this using different expectations µy and covariances Γy : Notes: For construction of conditional densities, see Theorem 4.4d.

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