0:00
All right, hello.
00:01
So here to show the sample mean is a consistent estimator of the population mean for our observations, x1, x2, to xn, that are independent and identically distributed.
00:11
That's what iid means.
00:12
We can use the properties of expectation and variance.
00:15
So we denote the sample mean as x bar, population mean as nu.
00:19
So the sample mean x bar is just equal to, we take 1 over n times the sum where i goes from 1 to n of x sub i.
00:30
And then the expectation of the sample mean, so first we show the expectation of the sample mean equals the population mean.
00:38
Since the xi's are iid, they have the same expectation mu.
00:46
So we get then that the expected value of e bar is going to be equal to just 1 over n times the sum i goes from 1 to n of e of x sub i, which is equal to just 1 over n times n times mu, which is equal to mu.
01:08
And so the sample mean is an unbiased estimator of the population mean.
01:14
Then we examine the variance of the sample mean...