5. (12 points) Consider a sample dataset with the binary response y that takes possible values of 0 and 1 and a single predictor r. We assume a follows a normal distribution with density function 1 1 fr(xly = k) = 12TO - exp 202 (X - HA) 2 ), where ux is the mean of the k-th class, k = 0, 1, o is the variance for both classes. The estimated parameter values for a training data are (1 = -2, po = 4, 62 = 3, #1 = 0.25, no = 0.75. We now use Linear Discriminant Analysis (LDA) to answer the following question: (a). Predict the posterior probability P(y = 1/x = 0). (b). Using your results from (a), make a prediction for the class for r = 0. (Use the default 0.5 as probability threshold for classification) Hint: The formulas for discriminant scores are: do() = x . 29 - 2 + log(no), 61(x) = x . 4 - 207 + log(#1). The posterior probability can be written as P(y = 1x) = simplified ebo (2) + ebi (2) Note: the function log() is the natural logarithmic function