Let ( X1, X2, ldots, Xn ) be a random sample from a normal...

Let ( X_{1}, X_{2}, ldots, X_{n} ) be a random sample from a normal distribution with mean ( mu ) and variance ( sigma^{2} ). 1. Derive the method-of-moments estimators of ( mu ) and ( sigma^{2} ); 2. Show that if the method-of-moments estimators of ( mu ) and ( sigma^{2} ) are unbiased estimators or not.

Transcript

00:01 Ok, so the method of moments is when we equate the theoretical mean with the sample mean or the sample expectations.

00:16 So we can see that for the sample, the expectation is defined as the sum of the x's.

00:24 And the expectation of x squared is defined as, sorry, times 1 over n, is defined as 1 over n times the sum of the squares of the x's.

00:42 And the theoretical values, the expectation of the x's is just mu and the expectation of the x squared is sigma squared plus mu squared.

00:55 And so if we equate these and set mu to mu hat to say it's our estimator and sigma to sigma hat to say it's our estimator, we find that we get that the method of moment estimator for mu hat is 1 over n times the sum of the x's.

01:15 And the estimator for sigma hat is 1 over n times the sum of the squares of the x's minus mu hat squared, or equivalently, 1 over n squared times the sum of all the x's squared.

01:47 And so then we're asked to, sorry, that was sigma squared.

01:53 So then we're asked to say whether these are unbiased or not.

01:58 And so to do that, we look at their expectation.

02:02 So the expectation of mu hat by the linearity of expectation is just 1 over n times the sum of the expectations of the x's.

02:10 They all have expectation mu and there's n of them.

02:13 So this is 1 over n times n mu, which is mu.

02:16 And therefore it is unbiased.

02:22 And then the expectation of sigma hat squared is just given by 1 over n times the sum of the expectations of the squares of the xi's...

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