00:01
So we have this probability density function f of x given theta, which is theta times x, where x is raised to the f of theta minus one power, and x is between zero and one, and theta is strictly greater than zero.
00:18
And we're given some data values as well, we'll talk about those in a second, x1, x1, x10 samples from this distribution, and what we're going to do is find the maximum likelihood estimator theta which is theta hat so let's go and do that so likelihood which is the product of our function one up to ten in this case normally the n if we don't know how many samples we're just trying to find the general maximum likelihood estimator but we have 10 samples and the product is theta x i and this is the variable we're going to iterate over x i so so let's see, if we think about what we have here, it's a bunch of these products here.
01:10
So it's 10 thetas multiplied together.
01:12
So it's theta to the 10th.
01:15
And then we have the product of all these xi's raised to the theta minus 1.
01:25
And we're kind of stuck right here as is.
01:29
So what we're going to do now, now, and that's because all these, these xis are the different faces.
01:34
So we can't really do anything with the exponents yet.
01:37
Like special exponent properties, like we'd likely do with theta.
01:40
But now we're gonna take the log of this likelihood function.
01:43
So log theta, and that's gonna help us maximize it.
01:47
So it's gonna be the log, natural log, theta to the 10th.
01:55
Now, it's these terms, we'll put it together, theta to the 10th times this product.
01:59
So we're gonna take the log, the sum of the logs then plus the natural log of, now we're gonna get something interesting.
02:07
So it's the log of this product of xi theta.
02:13
So this is gonna be where it gets kind of interesting.
02:18
Now we can use some properties of logarithms.
02:20
So the 10 will come out front, 10 natural log of theta.
02:23
So let's look at what we have here.
02:25
So this is a bunch of natural log.
02:30
It's gonna be xi to the theta minus one x1, excuse me, x1, right? so it's going to be 1 times x2 to the theta minus 1 times da -da -da all the way to x10 to the theta minus 1.
02:49
So now, again, we can use the properties of logs.
02:52
We're taking the log of this product.
02:55
It means we take the sum of the logs of those values...