00:01
Okay, so here we're given a random sample from a distribution with the probability density function given by f of x theta equal to theta x -mina minus 1 for x between 0 and 1 and 0.
00:24
Otherwise and we're asked to find an estimator for the parameter theta okay let me just pause first all right sorry i made a small break let's just continue okay so solution all right so here of course the first thing we have to do is to define our likelihood function of course this is defined as x1 xn as a parameter theta we shouldn't forget and this is defined, as you know, as the product i equals 1 to n of our function at xi.
01:09
We're considering a sample of n observations of the sample.
01:14
So let's just write with this.
01:17
Of course, we don't have to worry about the zero interval.
01:21
Let's just write the part that matters for us, which is that we will have the product of theta xi.
01:31
To the theta minus 1.
01:34
All right.
01:38
And let's just see what we, the theta can come out and we can think of it as being, say, the product of the x -i -theta -minus 1 from i equals 1 to n.
01:51
It's equivalent if you want to, of course, a to the theta, b to the theta, same as ab to the theta.
01:58
So we can write this in a sort of nicer way by having the product here, all the theta minus 1.
02:06
And we want to optimize this as a function of theta...