00:01
So in this question we have a length mu and an area mu squared.
00:09
And we have n measurements of mu, the set of xi from i equals 1 to n.
00:23
Now each measurement is unbiased, so the expectation value of each xi is going to be mu.
00:30
And it has a variance of sigma squared.
00:35
So now first of all, let's think about the mean.
00:38
X bar is going to be the sum x i over n from i equals 1 to n which means that the expected value of the mean is going to be the expected value of this sum and since the expected value is a linear operator it can go inside the sum which is the sum from i equals 1 to n because these are unbiased mu over n so the expected value of the mean is indeed mu now the variance of the mean is the variance of the sum from i equals 1 to n x i over n.
01:25
Now the variance is not a linear operator, but when the measurements are independent, we can still expand a sum.
01:37
But the variance of a random variable multiplied by something, if we want to take that thing outside, we have to square it.
01:50
But we know what the variance of each of these are.
01:52
They have variance sigma squared.
01:56
And that means that the variance of the mean is sigma squared...