5. Consider the following decision rule for a two-category decision problem where thefeature is a scalar r: Given a contant c, decide w, is the true state of nature if r > c,otherwise decide wr is the true state of nature.(a) Show that the probability of error for this rule is given byPE = P(wr) (2 lwi)dr + P(w2) S(x \w)dr(b) By differentiating the expression above, show that a necessary condition to min-imize Pr is that c satisfyS(clwr)P(w) = f(clws)P(w2)What optimality criterion does this condition imply? (ie. Bayes optimal, Maxi-mum Likelihood, etc.)

The problem can be solved using the concept of decision rule and integration. please find step by…

5. Consider the following decision rule for a two-category decision problem where the feature is a scalar r: Given a contant c, decide w, is the true state of nature if r > e, otherwise decide wz is the true state of nature. (a) Show that the probability of error for this rule is given by PR = P(ur) (rlwi)dr + P(w2) S(r w)dr (b) By differentiating the expression above, show that a necessary condition to min- imize Pr is that e satisfy S(clun )P(w) = S(clws)P(w) What optimality criterion does this condition imply? (ie. Bayes optimal, Maxi- mum Likelihood, etc.)

5. Consider the following decision rule for a two-category decision problem where the feature is a scalar r: Given a contant c, decide w, is the true state of nature if r > e, otherwise decide wz is the true state of nature. (a) Show that the probability of error for this rule is given by PR = P(ur) (rlwi)dr + P(w2) S(r w)dr (b) By differentiating the expression above, show that a necessary condition to min- imize Pr is that e satisfy S(clun )P(w) = S(clws)P(w) What optimality criterion does this condition imply? (ie. Bayes optimal, Maxi- mum Likelihood, etc.)

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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