Question

Assume a gaussian-noise communication channel where the signal is either a 5 or a 0. The probability of either outcome is the same. Using the MAP criterion, regardless of the noise variance, we choose the alternative hypothesis if the test statistic y=?

Answer 1

To use the MAP criterion, we choose the alternative hypothesis if the likelihood ratio is greater than or equal to 1.

Step-by-step explanation:

To use the MAP criterion, we choose the alternative hypothesis if the likelihood ratio is greater than or equal to 1. The likelihood ratio is the probability of the data given the alternative hypothesis divided by the probability of the data given the null hypothesis.

In this case, the data is either a 5 or a 0, and the probability of either outcome is the same. So the probability of the data given the null hypothesis is 0.5, and the probability of the data given the alternative hypothesis is also 0.5.

Therefore, the likelihood ratio is 0.5/0.5 = 1. Since 1 is not greater than or equal to 1, we would not choose the alternative hypothesis. We would choose the null hypothesis instead.