00:01
So in this question we have y1 up to yn.
00:06
A random sample from f of y is theta y to the theta minus 1 for y between 0 and 1 and 0 elsewhere.
00:24
So let's get the expectation of variance of y.
00:27
The expectation of y is the integral from 0 to 1 theta y to the theta dy, which is theta over theta plus 1 y to the theta plus 1 evaluated between 0 and 1.
00:39
So it's theta over theta plus 1.
00:41
Now we need the expectation of y squared because we're going to need its variance to do this question as well.
00:54
And that gives us theta over theta plus 2 y to the theta plus 2 from 0 to 1, which is theta over theta plus 2.
01:03
So now the variance of y is the expectation of y squared minus the expectation of y squared.
01:14
So that's theta over theta plus 2 minus theta squared over theta plus 1 squared, which is theta theta plus 1 squared minus theta squared theta plus 2 divided by theta plus 1 squared theta plus 2.
01:37
So we can pull out a theta, and then we've got theta plus 1 squared, theta squared plus 2, theta plus 1, minus theta times theta plus 2.
01:46
So minus theta squared minus 2 theta.
01:49
They cancel, so we're just left with the variance of theta, sorry, the variance of y is theta over theta plus 1 squared theta plus 2.
02:06
And let's recap the expectation.
02:08
The expectation of y was theta over theta plus 1.
02:16
Right, so now we've got that.
02:18
Let's think about y bar.
02:21
Y bar is 1 over n times the sum from 1 to n of y's.
02:26
So that means that the expected value of y bar is 1 over n times the sum from 1 to n.
02:33
The expected value of the y's, which we can go from here, is theta over theta plus 1.
02:39
But that just means the n's cancel out, and we get the expected value is theta over theta plus 1.
02:45
Now the variance of y bar is going to be 1 over n squared times the sum of the variances of the y's, which is 1 over n times theta over theta plus 1 squared theta plus 2...