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Problem 1 - Linear Discriminant Analysis: Consider the categorical learning problem con- sisting of a data set with two labels: Label 1: X1 3.81 0.23 3.05 0.68 2.67 X2 -0.55 3.37 3.53 1.84 2.74 Label 2: X1 -2.04 -0.72 -2.46 -3.51 -2.05 X2 -1.25 -3.35 -1.31 0.13 -2.82 a) For each label above, the data follow a multivariate normal distribution N(Hi, ), where the covariance is the same for both label 1 and for label 2. Fit a pair of Guassian discriminant func- tions to the labels by computing the covariances, means, and proportions of datapoints as discussed in the Linear Discriminant Analysis section of Lecture 5. You may use a computer, but you should not use an LDA solver. You should report the values for Hi and . b) Give the formula for the line forming the discretion boundary.

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# (a)

x1 <- matrix(c(3.81, 0.23, 3.05, 0.68, 2.67,
             -0.55, 3.37, 3.53, 1.84, 2.74),
               5, 2)

x2 <- matrix(c(-2.04, -0.72, -2.46, -3.51, -2.05,
             -1.25, -3.35, -1.31, 0.13, -2.82),
            5, 2)


pi1 <- pi2 <- 5/10
mu1 <- colMeans(x1)
# [1] 2.088 2.186
mu2 <- colMeans(x2)
# [1] -2.156 -1.720...

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