hw3_sol.pdf - CS189Spring 2016 Solutions to Homework 3...

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CS189–Spring 2016 — Solutions to Homework 3Shaun Singh,TAMarch 13, 2016Problem 1: Independence vs. Correlation(a) Essentially, there are 4 possible points (X, Y) can be, all with equal probability (14):{(0,1),(0,-1),(1,0),(-1,0)}, If graphed onto the Cartesian Plane, these point form”crosshairs”.To show that X and Y are uncorrelated, we need to prove:
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CS189–Spring 2016 Homework 3 Shaun Singh,TA2
CS189–Spring 2016 Homework 3 Shaun Singh,TA3Problem 2: Isocontours of Normal Distributions(a) See appendix for graphs
CS189–Spring 2016 Homework 3 Shaun Singh,TA4Problem 3: Visualizing Eigenvectors of Gaussian CovarianceMatrix(a) See appendix for graphs
CS189–Spring 2016 Homework 3 Shaun Singh,TA5Problem 4: Covariance Matrices and Decompositions(a) We know that without loss of generality that the covariance matrix ΣXRN,Ncor-responding to random variableXRNis positive semidefinite, which means thatxRN,x>Σx0. Unfortunately, we require that Σ be positive definite (xRN,x>Σx >0) in order for it to be invertible, since invertible matrices cannot have anyeigenvalues be 0.When might our covariance matrix Σ have an eigenvalue of 0? The most general case

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