00:01
So we're told x is a discrete random variable with the following probability distribution function.
00:07
We have x can be 0, 1, 2, or 3.
00:10
And we have 2 pi over 3 for 0, or excuse me, 2 theta over 3 for 0, theta over 3 for 1, 2 times 1 minus theta over 3 for 2, and 2 times 1 minus theta over 3 for 3.
00:32
And theta is between 0 and 1.
00:35
And what we are trying to do today is to figure out the maximum likelihood estimate of theta, the mle of theta, which would be theta hat.
00:50
And so we're going to use the likelihood function, which we get.
00:58
So capital l of theta is the product iterating from i is 1 up to n of our function here, but at these different values.
01:12
So whatever probability of x is 0, probability x is 1, probability x is 2, and probability x is 3.
01:26
And we're given some data, actually.
01:28
We're given 3, 0, 2, 1, 3, 2, 1, 0, 2, and 1.
01:38
So 10 data points.
01:39
1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
01:42
So minus 10.
01:45
And so how we use this is we recognize that, ok, how many 0s were there? there's one 0 and two 0s.
01:52
So that means probability of 0, this happened twice.
01:55
So we can get 2 theta over 3 squared.
02:02
How many times did 1 occur? we got rid of the 0s.
02:04
1 occurred 3 times, right? 1, 2, 3.
02:08
That means 1 occurred 3 times.
02:11
So we've got theta over 3 to the third.
02:14
And how many times did we get 2? 1, 2, and 3.
02:19
So we got that 3 times, so 2 times 1 over theta, or 1 minus theta over 3 cubed.
02:27
And then 3 also went twice, right? those two values.
02:31
So we get 2 times 1 minus theta over 3 all to the second power.
02:43
All right, so now we multiply all these together.
02:47
Actually, we don't need the product anymore.
02:49
That's it.
02:49
We have it.
02:50
That's our product.
02:51
And we don't need to worry about the constants.
02:54
So we're just going to kind of ignore the constants.
02:57
All right, so it's 2 thirds to the second...