binan EXERCISE 2.1 (Hypothesis testing: Uniform and uniform) Consider a hypothesis testing problem in which the hypotheses H = 0 and H = 1 occur probability P (0) and P (1) 1 - PH(O), respectively. The observable y take values in {0,1}2, where k is a fixed positive integer. When H = 0, each componen of Y is 0 or 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 thai have an equal number of ones and zeros. There are (26) such sequences. (a) What is Pyin (y|0)? What is Pyju(41)? (b) Find a maximum-likelihood decision rule for H based on y. 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) such that the decision rule that minimizes the erro, probability always chooses the same hypothesis regardless of y? If yes, what are these values, and what is the decision? binan EXERCISE 2.1 (Hypothesis testing: Uniform and uniform) Consider a hypothesis testing problem in which the hypotheses H = 0 and H = 1 occur probability P (0) and P (1) 1 - PH(O), respectively. The observable y take values in {0,1}2, where k is a fixed positive integer. When H = 0, each componen of Y is 0 or 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 thai have an equal number of ones and zeros. There are (26) such sequences. (a) What is Pyin (y|0)? What is Pyju(41)? (b) Find a maximum-likelihood decision rule for H based on y. 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) such that the decision rule that minimizes the erro, probability always chooses the same hypothesis regardless of y? If yes, what are these values, and what is the decision