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statistical_sampling

# statistical_sampling - Estimation of Parameters and...

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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 σ 2 , but do not know the mean μ . One way to estimate μ 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 ˆ μ = X 1 + ··· + X k k be an estimator for μ . Note that ˆ μ is a random variable, and for large values of k it will have approximately a normal distribution with mean μ (by the Central Limit Theorem). The sort of thing we would like to compute is a 95% confidence interval for μ , which is an interval (ˆ μ- δ, ˆ μ + δ ) such that 95% of the time (remember, ˆ μ is a random variable), μ 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 σ , but not μ . Problem 2 (real). This is the exact same problem, except that here we know neither μ nor σ ; 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 ( μ, σ 2 ), and we consider ˆ μ = X 1 + ··· + X k k and ˆ σ 2 = 1 k- 1 k summationdisplay i =1 ( X i- X ) 2 . The problem here is to determine δ such that (ˆ μ- δ, ˆ μ + δ ) is a 95% confidence interval for μ ; and, we would furthermore like a 95% confidence interval for σ 2 (or just σ )....
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statistical_sampling - Estimation of Parameters and...

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