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Find the probability of error

A pixel in a particular image has a random brightness, X. The distribution of X depends on whether the pixel belongs to an object (O) of interest or the background (B). If it belongs to the object, then X has a Laplace distribution, fX(x|O) = ? 2 exp(??|x|), ?? < x < ?, where ? = 0.2. If the pixel belongs to the background, then X is uniformly distributed between -10 and 10. Consider the problem of classifying a pixel based on an observation of its brightness, X = x. (a) Find the ML rule for all values of x and illustrate the decision regions on a number line. (b) Find the probability of error if the ML decision rule is used, but the probability that the pixel belongs to the object of interest is equal to 0.6. (c) Adjacent pixels can give information that can be treated as a priori probabilities in determining whether the pixel belongs to the background or the object. If information from adjacent pixels implies that this pixel should belong to the object with probability 3/4, find the MAP decision rule given X = x for all values of x. (d) If information from adjacent pixels implies that this pixel should belong to the object with probability 0.2, find the MAP decision rule given X = x for all values of x.

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