Need some HW help on Exercise 2.4

EXERCISE 2.4 (Poisson parameter estimation) In this example there are two hypotheses, H = 0 and H = 1, which occur with probabilities Ph(0) = po and Ph(1) = 1 Po, respectively. The observable Y takes values in the set of non- negative integers. Under hypothesis H = 0, Y is distributed according to a Poisson law with parameter 1o, i.e. 1 Py\h(y|0) = (2.35) y! -e-to Under hypothesis H = 1, Pyh(y|1) = 11 (2.36) y! This is a model for the reception of photons in optical communication. (a) Derive the MAP decision rule by indicating likelihood and log likelihood ratios. Hint: The direction of an inequality changes if both sides are multiplied by a negative number. (b) Derive an expression for the probability of error of the MAP decision rule. (c) For Po = - 1/3, lo = 2 and li 10, compute the probability of error of the MAP decision rule. You may want to use a computer program to do this. (d) Repeat (c) with 11 = 20 and comment. EXERCISE 2.5 (Lie detector) You are asked to develop a "lie detector" and analyze its performance. Based on the observation of brain-cell activity, your EXERCISE 2.4 (Poisson parameter estimation) In this example there are two hypotheses, H = 0 and H = 1, which occur with probabilities Ph(0) = po and Ph(1) = 1 Po, respectively. The observable Y takes values in the set of non- negative integers. Under hypothesis H = 0, Y is distributed according to a Poisson law with parameter 1o, i.e. 1 Py\h(y|0) = (2.35) y! -e-to Under hypothesis H = 1, Pyh(y|1) = 11 (2.36) y! This is a model for the reception of photons in optical communication. (a) Derive the MAP decision rule by indicating likelihood and log likelihood ratios. Hint: The direction of an inequality changes if both sides are multiplied by a negative number. (b) Derive an expression for the probability of error of the MAP decision rule. (c) For Po = - 1/3, lo = 2 and li 10, compute the probability of error of the MAP decision rule. You may want to use a computer program to do this. (d) Repeat (c) with 11 = 20 and comment. EXERCISE 2.5 (Lie detector) You are asked to develop a "lie detector" and analyze its performance. Based on the observation of brain-cell activity, your