Reading:
EECE 5644 Homework #1
Appendices A.1 – A.5, Notes, Chapter 2.1–2.7 Let x be a real-valued random variable.
- (a) Prove that the variance of x = 2 = E[(x μ)2] = E[x2] μ2.
- (b) Let x be a real-valued random vector. Prove that the covariance
matrix of x = ⌃ = E[xxT ] μμT .
Suppose two equally probable one-dimensional densities are of the form
p(x|!i) / e|xai|/bi for i = 1,2 and b > 0.
- (a) Write an analytic expression for each density, that is, normalize each
function for arbitrary ai, and positive bi.
- (b) Calculate the likelihood ratio p(x|!1)/p(x|!2) as a function of your
four variables.
- (c) Plot a graph (using MATLAB) of the likelihood ratio for the case a1 =0,b1 =1,a2 =1andb2 =2. Makesuretheplotsarecorrectly labeled (axis, titles, legend, etc) and that the fonts are legible when printed.
Consider a two-class problem, with classes c1 and c2 where P (c1) = P(c2) = 0.5. 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.)
Consider a two-class, one-dimensional problem where P(!1) = P(!2) and p(x|!i)⇠N(μi,i2). Letμ1 =0,12 =1,μ2 =μ,and2 =2.
- (a) Derive a general expression for the location of the Bayes optimal decision boundary as a function of μ and 2.
- (b) 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.
- (c) Estimate the Bayes error rate pe.
- (d) 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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