This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Estimation of Parameters and Statistical Sampling November 6, 2010 1 Introduction Here we consider two types of statistical sampling problems, one is just for pedagogical purposes, the other is directly applicable to real problems. These two problems are: Problem 1 (pedagogical). Suppose that X is a random variable for which we know the variance 2 , but do not know the mean . One way to estimate would be to take samples of X , and then average. That is, suppose that X 1 , ..., X k are independent random variables with the same distribution as X ; then, we let = X 1 + + X k k be an estimator for . Note that is a random variable, and for large values of k it will have approximately a normal distribution with mean (by the Central Limit Theorem). The sort of thing we would like to compute is a 95% confidence interval for , which is an interval ( - , + ) such that 95% of the time (remember, is a random variable), lies in this interval. The reason that this problem is only pedagogical is that in real world problems we are unlikely to encounter situations where we know , but not . Problem 2 (real). This is the exact same problem, except that here we know neither nor ; in addition , we will assume that X is normal (a 1 standard assumption for many statistical sampling problems). This problem is vastly more difficult to analyze theoretically; however, we are in luck that it was worked out long ago. There is actually a nice little bit of history surrounding this that we will discuss below. Basically, as before, we suppose that X 1 , ..., X k are independent and have the same distribution as X = N ( , 2 ), and we consider = X 1 + + X k k and 2 = 1 k- 1 k summationdisplay i =1 ( X i- X ) 2 . The problem here is to determine such that ( - , + ) is a 95% confidence interval for ; and, we would furthermore like a 95% confidence interval for 2 (or just )....
View Full Document