The class-conditional probability density functions (pdf) of a feature x are shown in the figure below: p(x|w1) h p(x|w2) 0 1 2 3 4 5 X 6 7 8 9 10 The prior probabilities of the two classes are P(w1) = 2/3 and P(w2) = 1/3. Do the followings: a) (2 pts) Determine the value of h. b) (2 pts) Sketch the pdfs p(x, w1) and p(x,w2) in one figure. c) (2 pts) Sketch the evidence p(x). d) (2 pts) Sketch the posterior probability distributions P(w1|x) and P(w2|x) in one figure. e) (6 pts) For the minimum error classifier, determine the decision regions for the two classes w1 and w2. f) (6 pts) Compute the Bayes error P(error) (note: it is easier to compute the Bayes error based on the figure produced in part b).
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Let's call the pdfs p(x|w1) and p(x|w2). We need to solve the equation p(x|w1) = p(x|w2) for x. Show more…
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(a) Assume there are 2 classes, w1 and w2, and one feature vector x, give the Bayes rule for classification in terms of a priori probabilities, P(w1) and P(w2), and the class-conditional probability densities of x. (b) Suppose we have a two-classes problem (A, B), with a single binary-valued feature x in {0, 1}. Assume the prior probability P(A) = 0.4. Given the distribution of the samples as shown in the following table, use Bayes Rule to compute the values of the four posterior probabilities: P(A|x = 1), P(A|x = 0), P(B|x = 1), and P(B|x = 0).
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The article "Error Distribution in Navigation" (J. Institut. Navigation, $1971 : 429-442$ ) suggests that the frequency distribution of positive errors (magnitudes of errors) is well approximated by an exponential distribution. Let $X=$ the lateral position error (nautical miles), which can be either negative or positive. Suppose the pdf of $X$ is $$f(x)=.1 e^{-2|x|} \quad-\infty< x<\infty$$ (a) Sketch a graph of $f(x)$ and verify that $f(x)$ is a legitimate pdf (show that it integrates to 1$)$ . (b) Obtain the cdf of $X$ and sketch it. (c) Compute $P(X \leq 0), P(X \leq 2), P(-1 \leq X \leq 2),$ and the probability that an error of more than 2 miles is made.
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Supplementary Exercises
Let the random variables X and Y have a joint probability density function (pdf) fX,Y(x,y) = cxy^2, where 0 < y < x < 2. (a) Find the value of c that makes fX,Y(x,y) a valid pdf. [3] (b) Calculate the marginal density functions for X and Y. [4] (c) Find the conditional density function of Y | X. [2] (d) Calculate E(X) and E(Y | X). [4] (e) Determine whether X and Y are independent or not.
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