00:01
So in this question, we're told that the probability of the gene a -a is theta -squared.
00:09
The probability of the mixed gene is 2 -theta -1 -1 -theta, and the probability of the gene double big -a is 1 -theta.
00:23
Squared, and we observe n -a -n -a -n -a -n -a -n -a -n -a observed.
00:35
So the likelihood function of theta, given these three pieces of information, is n -factorial, so that's the sum of all of these three, divided by n -a -a -a -factorial, n -a -a -factorial, n -a -a -factorial.
01:00
And then we're going to have theta to the two naa.
01:06
Basically, we have the product of the probabilities.
01:10
So p -a -n -a, etc.
01:22
So our log likelihood function is going to be, so we just take the log of this.
01:37
So i'm going to call this thing, this is going to be log of n -n -n -a, n -a, n -a -n -a, and then we add.
01:51
Now log paa, paa is theta squared, so this is going to give us 2 log theta.
02:03
The mixed probability is 2 theta 1 minus theta.
02:07
So this is going to give us naa log 2 theta 1 minus theta.
02:16
And then here we're going to get big naa times 2 1 minus theta.
02:29
So now we want to find the maximum likelihood estimator.
02:33
And for that, the maximum likelihood estimator is given b by d theta of the log likelihood.
02:48
Evaluated at the maximum likelihood estimator theta hat is equal to 0.
02:53
So 0 equals, so let's take a derivative of this, 2n lill -a -l -a over theta...