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Problem 3 Let Xl, X2, ..., Xn be an i.i.d. sample and Fn(x) the empirical distribution function. We know from the central limit theorem that- 4 MCO, 02) with 02 — Fx(x)) depending on x. Hence, for large n the distribution of Fn(x) will be close to a normal distribution. (a) (2 marks) Find an upper bound for th...

Problem 1 (a) (2 marks) Let X be random variables. Check that Var(X) E(X2) — (b) (2 marks) Let X, y be random variables. Check that var(x + Y) Var(X) + Var(Y) + 2Cov(x, Y). Problem 2 (a) (5 marks)...

3. Consider a binary classification problem Y G {0, 1} with one predictor X. The prior probability of being in class 0 is Pr(Y = 0) function for X in class 0 is a standard normal. 2 The distribution for X in class 1 is the exponential distribution with parameter A 1/2. The density function is
(a) Plot Tofo(x) and Tlfl(x) in the same figure.

(b) Find the Bayes decision boundary (Hint: rof...