(a) MAP hypothesis testing rules: P(H_i | N) = P(N | H_i) * P(H_i) / P(N)
b) P(10 | H_1) > P(10 | H_0), the ML rule would also choose hypothesis H_1.
The problem is as follows:
In a random hour, the number of call attempts N at a telephone switch has a Poisson distribution, either with a mean of α_0 = 8 (hypothesis H_0) or α_1 = 20 (hypothesis H_1).
The a priori probabilities are P(H_0) = 0.9 and P(H_1) = 0.1.
We need to find the MAP and ML hypothesis testing rules given the observation of N.
MAP (Maximum A Posteriori) Hypothesis Testing Rule:
The MAP rule maximizes the posterior probability P(H | N), which can be written as:
P(H | N) = P(N | H) * P(H) / P(N)
where:
P(N | H) is the likelihood of observing N given hypothesis H.
P(H) is the a priori probability of hypothesis H.
P(N) is the marginal probability of observing N, which can be obtained by summing P(N | H) * P(H) over all possible values of H.
For this problem, the likelihood function for a Poisson distribution is:
P(N | H) = e^(-λ) * λ^N / N!
where λ is the mean of the Poisson distribution under hypothesis H.
Therefore, we can calculate the posterior probability for each hypothesis and choose the hypothesis with the higher posterior probability:
P(H_0 | N) = (e^(-8) * 8^N / N!) * 0.9 / P(N)
P(H_1 | N) = (e^(-20) * 20^N / N!) * 0.1 / P(N)
The decision rule is then:
N ∈ A_0 if P(H_0 | N) > P(H_1 | N)
N ∈ A_1 otherwise
ML (Maximum Likelihood) Hypothesis Testing Rule:
The ML rule simply chooses the hypothesis that maximizes the likelihood P(N | H):
N ∈ A_0 if P(N | H_0) > P(N | H_1)
N ∈ A_1 otherwise
Example:
Suppose we observe N = 10 call attempts.
Then, we can calculate the posterior probabilities:
P(H_0 | 10) ≈ 0.046
P(H_1 | 10) ≈ 0.147
Since P(H_1 | 10) > P(H_0 | 10), the MAP rule would choose hypothesis H_1.
Similarly, we can calculate the likelihoods:
P(10 | H_0) ≈ 0.022
P(10 | H_1) ≈ 0.180
Since P(10 | H_1) > P(10 | H_0), the ML rule would also choose hypothesis H_1.
Therefore, for this example, both the MAP and ML rules would conclude that the number of call attempts is more likely to have come from a Poisson distribution with a mean of 20 (hypothesis H_1).