Description
Reading: Appendices A.1 – A.5, Notes, Chapter 2.1–2.7
1.1 (10 pts) Let x be a real-valued random variable.
- Prove that the variance of x = 2 = E[(x µ)2] = E[x2] µ2.
- Let x be a real-valued random vector. Prove that the covariance matrix of x = ⌃ = E[xxT] µµT.
1.2 (10 pts) Suppose two equally probable one-dimensional densities are of the form p(x|!i) / e |x ai|/bi for i = 1,2 and b > 0.
- Write an analytic expression for each density, that is, normalize eachfunction for arbitrary ai, and positive bi.
- Calculate the likelihood ratio p(x|!1)/p(x|!2) as a function of your four variables.
- Plot a graph (using MATLAB) of the likelihood ratio for the casea1 = 0, b1 = 1, a2 = 1 and b2 = 2. Make sure the plots are correctly labeled (axis, titles, legend, etc) and that the fonts are legible when printed.
- (10 pts) Consider a two-class problem, with classes c1 and c2 where P(c1) = P(c2) = 0. There is a one-dimensional feature variable x. Assume that the x data for class one is uniformly distributed between a and b, and the x data for class two is uniformly distributed between r and t. Assume that a < r < b < t. Derive a general expression for the Bayes error rate for this problem. (Hint: a sketch may help you think about the solution.)
- (12 pts) Consider a two-class, one-dimensional problem where P(!1) = P(!2) and p(x|!i) ⇠ N(µi, i2). Let µ1 = 0, 12 = 1, µ2 = µ, and 22 = 2.
- Derive a general expression for the location of the Bayes optimaldecision boundary as a function of µ and 2.
- With µ = 1 and 2 = 2, make two plots using MATLAB: one for the class conditional pdfs p(x|!i) and one for the posterior probabilities p(!i|x) with the location of the optimal decision regions. Make sure the plots are correctly labeled (axis, titles, legend, etc) and that the fonts are legible when printed.
- Estimate the Bayes error rate pe.
- Comment on the case where µ = 0, and 2 is much greater than 1. Describe a practical example of a pattern classification problem where such a situation might arise.
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