Final answer:
To determine the MAP test, calculate the likelihood ratio for H₀ and H₁, compare it with the decision threshold based on prior probabilities, and reject H₀ if the ratio exceeds the threshold.
Step-by-step explanation:
To find the Maximum A posteriori Probability (MAP) test for deciding from which of two possible probability distributions a sample comes from, when H₀: x∼N(0,1) and H₁: x∼N(0,2), start by calculating the likelihood ratio for the two hypotheses.
Given that the prior probabilities P(H₀) and P(H₁) are known, the test statistic is the likelihood ratio, and we compare it to a decision threshold based on the prior probabilities.
Calculate the likelihood of observing the data under each hypothesis.
Form the likelihood ratio by dividing the likelihood under H₁ by that under H₀.
The decision threshold is derived from the ratio of the prior probabilities. For example, when P(H₀) = 1/2, the threshold is 1, and when P(H₀) = 3/4, the threshold is 3 because P(H₁) = 1 - P(H₀).
If the likelihood ratio exceeds the threshold, reject H₀ in favor of H₁; otherwise, do not reject H₀.
The normal distribution is used in this scenario because we assume we know the population standard deviation or have a sufficiently large sample size.
Decision thresholds vary depending on the prior probability of each hypothesis, affecting whether we reject the null hypothesis or not.