00:01
So in this question we want to calculate a likelihood ratio test statistic.
00:05
So we have the probability that x -i equals k is p 1 -p to the k -minus 1 for k -2, etc.
00:14
And we have x, and we have i is equal to 1 to n.
00:23
So we have h -naut is that p is a half.
00:32
So and h1 is that p is not equal to a half.
00:36
So that means that theta nought is the set of p where p is equal to a half, and theta is the set of p where p is between 0 and 1.
00:59
And these are our sets of parameters that we're going to be maximizing the likelihood over.
01:05
So the likelihood of theta given x, so we have theta is p in both cases.
01:14
The likelihood of theta given our vector of data x is p to the n, 1 minus p to the sum of xx of xi minus n because we just multiply it altogether.
01:35
So lambda x is the maximum over our set theta nought of the likelihood divided by the maximum over our full set theta of the likelihood.
01:58
So, and then the log of that, the log of our likelihood ratio is the log of this quotient, which is the same as the difference of two logs.
02:07
So it's the log of the maximum over theta naught of the likelihood, minus the log of the maximum over theta of the likelihood.
02:25
But since log is monotonic, we can bring the maximum outside the log.
02:29
Max theta naught minus max theta, of the log likelihood.
02:40
So what's the log likelihood? the log likelihood is just the log of this function here.
02:47
So this is the max over p equals a half, minus the max over p in 0 to 1, of the log likelihood.
03:00
So let's write down what that is...