Question

In a random hour, the number of call attempts N at a telephone switch has a Poisson distribution, either with a mean of α0​=9 (hypothesis H0​) or α1​=19( hypothesis H1​). Assume a priori probabilities P[H0​]=0.8 and P[H1​]=0.2. Note: Your answers below must be integers. (a) Find MAP hypothesis testing rules given the observation of N. n∈A0​ if n=0,1,…,;n∈A1​ otherwise (b) Find ML hypothesis testing rules given the observation of N. n∈A0​ if n=0,1,…,;n∈A1​ otherwise

Answer 1

(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.

Answer 2

This is a Poisson distribution problem involving MAP and ML hypothesis testing rules to decide between two mean call number hypotheses, factoring in both prior probabilities for MAP and likelihood for ML.

Step-by-step explanation:

The student's question involves hypothesis testing with Poisson distributions in which there are two hypotheses H0 and H1, characterizing two different values for the mean number of calls (α0=9 and α1=19 respectively). The MAP hypothesis testing rule and the ML hypothesis testing rule are needed to decide between these hypotheses based on the observed number of calls N.

For the MAP (Maximum A Posteriori) testing rule, the decision is based not only on the likelihood of observing N given each hypothesis but also takes into account the prior probabilities P[H0] and P[H1]. The MAP decision rule favors the hypothesis that has the highest posterior probability for the observed N.

The ML (Maximum Likelihood) testing rule, on the other hand, only considers the likelihood of observing N under each hypothesis and chooses the hypothesis under which the observation is most likely. Therefore, the value of N would determine whether n falls within A0 (favoring H0) or A1 (favoring H1) using the likelihood ratio.