Final answer:
(a) MAP hypothesis testing rules: Choose hypothesis H0 if N ≤ 11, else choose hypothesis H1.
(b) ML hypothesis testing rules: Choose hypothesis H0 if N = 0, else choose hypothesis H1.
Step-by-step explanation:
For Maximum A Posteriori (MAP) hypothesis testing, the rule is to select the hypothesis that maximizes the posterior probability given the observation. Given a Poisson distribution, MAP testing compares the probabilities based on the observed number of calls. Considering P[H0] = 0.8 and P[H1] = 0.2, the MAP rule is to select H0 if the observed number of calls, N, is less than or equal to the threshold value obtained from α0 (mean = 9) corresponding to the cumulative probability of 0.8, which is around 11. Therefore, if N is less than or equal to 11, hypothesis H0 is chosen; otherwise, hypothesis H1 is selected.
Maximum Likelihood (ML) hypothesis testing involves choosing the hypothesis that maximizes the likelihood of the observed data. In this case, given N as the number of call attempts, the ML rule for a Poisson distribution is to select H0 if N = 0 (as it maximizes the likelihood for α0 = 9), as it's the most probable outcome under this hypothesis.
If N is any value other than zero, the ML rule suggests choosing H1 because α1 = 19 would provide a higher likelihood for generating larger values of N compared to α0 = 9. Therefore, the ML hypothesis testing rule states to choose H0 if N = 0, and H1 otherwise.