Raul Miller wrote: > On 6/29/07, John Randall <[EMAIL PROTECTED]> wrote:
> But I believe n(\sigma^2/n) is \sigma^2 and I cannot find any > interpretation of $E(S^2)=(1/n-1)(n\sigma^2 -(\sigma^2)) that > seems numerically valid. > ... This is a statement about the expected value of a random variable. You cannot numerically validate it except in cases where the sample space is finite and you know the distribution. However, the above statement is asserting that S^2 is an unbiased estimator of \sigma^2 for any distribution. . > > It seems to me that \sigma^2(X_i) is 0.25 which means I should expect > that \sigma^2/n is 0.125. But from previous calculations, I expect that > \sigma^2(bar X) is 0.125 which means that either I have woefully > misunderstood some of the notation, or that the assertion > $\sigma^2(\bar X)=\sigma^2/n$ > > must be false. Your calculations seem entirely consistent. What is the problem. >> >> What is a sample which does not represent the population? >> > >> > The issue seems to be completeness. >> > >> > The "unbiased" estimator traditionally gets used when dealing with >> > variance for a sample which is not identical to the population. In >> > other words, if there's any possibility that the distribution of the >> > sample is different from the distribution of the population, it seems >> > to be traditional to use the "unbiased estimator" (n/n-1 when >> determining >> > RMS deviation rather than the "biased estimator" of 1 when determining >> > RMS deviation). >> >> I don't understand this. A random sample is defined to be a list of >> independent random variables each with the same distribution as the >> population. What does "a sample which is not identical to the >> population" >> mean? > > The distribution of the random variables which were used to generate > the sample is not, in the general case, the same as the distribution > of an instance of a sample. > > For example, let's consider our simple case: we count the number > of times "heads" comes up, when we flip a fair coin. > > The distribution of the population is: > 0: 50% > 1: 50% > > However, any single random sample containing three values > cannot represent this distribution. The potential samples are: > 0 0 0 (0: 100%, 1: 0%) > 0 0 1 (0: 66.67% 1: 33.33%) > etc. > A single random variable containing one value can "represent" this population: counting the number of heads when we flip a coin is precisely this. A particular random sample just has some distribution: in this way it is representative of the population. Best wishes, John ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
