Question

Problem 6 Hypothesis Testing: Uniform and Uniform) Consider a binary hypothesis testing problem in which the hypotheses H 0 and H 1 occur with probability PH (0) and PH (1) 1-PH (0), respectively. The observation Y is a sequence of zeros and ones of length 2k, where k is a fixed integer. When H 0, each component of Y is 0 or a 1 with probability and components are independent, when H = 1, Y is chosen uniformly at random from the set of all sequences of length 2k that have an equal number of ones and zeros. There are (z such sequences 2k (a) What is Pr H(0)? What is PrH(v1)? (b) Find a maximum likelihood decision rule. What is the single number you need to know about y to implement this decision rule? (c) Find a decision rule that minimizes the error probability. (d) Are there values of PH(0) and PH(1) such that the decision rule that minimizes the error probability always decides for only one of the alternatives? If yes, what are these values, and what is the decision?

Answer 1

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