Specifically, you are required to generate and plot 2-dimensional Gaussian random vector with z mean vector and the covariance matrix given by 1 p k for k = 1,2,.,9. 10 where p=

What is the covariance matrix V for M = 3 independent experiments with means m1, m2, m3 and variances a-r, (T�, O"§ ?

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1) Assuming you have a data matrix X that has n rows and p variables and you know both µ and Σ. How is (X- µ)‘Σ-1(X- µ) distributed? 2) Assuming that you don’t know the values of µ and Σ. How is the statistical distance distributed as n-p gets large?

(b) Compute the covariance matrix of the data given by x1 = [2 1]', x2 = [3 2]', x3 = [2 3]'and x4 = [1 4]'.

Please solve part (a)correctly in half hour . Please provide neat and clean work

You are given a dataset with n samples and three features. The covariance matrix C has eigenvalues (A1, A2, A3) = (5,3, 1) and associated normalized eigenvectors (vị, V2, V3), ||v;|| = 1. Check the correct statements: Vj • Vk = Ski O b. v Cv3 = 0 maxw w"Cw = 9 where w e R°and||w|| = 1 O d. v Cv, = 3 a. Oc. O e. wCw >0 where w e R

The covariance-variance matrix of a set of observations, xx, is a symmetric matrix. The Covariance matrix of functions of these observations should also be symmetric as the two covariance matrices are related by the equation Eyy= JEXXJT. Using laws on matrix transposition, prove that, indeed, Eyy is symmetric by showing that the product JEXXJT is symmetric, i.e., the original product and its transpose are equal. You need to list the transposition law concerned, or a property of a matrix, at every step of the proof.

1. If y = (y1-y2-Y,) be a random vector with mean vector and covariance matrix (1 1 0 E=|1 2 3 0 3 10 H= -1 3 Apply the appropriate matrices and linear algebra concepts to determine parameters of the distribution for (a) z when z= 2y, – 3y, + Y3 - (b) z= (z.2,) with z, = y + Y2 + y3 and Z, = 3y; + Y2 – 2y; -

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