On 6/30/07, John Randall <[EMAIL PROTECTED]> wrote: Lots of stuff, starting with:
The standard terminology covers all these cases, although there are some ambiguities. First note that a sample space is just a set of outcomes: is is different from a sample, which is a list of random variables.
First off, I should note that your terminology as you have outlined it here seems entirely consistent, though there are a few subtleties (such as, in this quoted paragraph, the distinction between sample space (which roughly corresponds to a result of the dyad /:[EMAIL PROTECTED]) and sample (which roughly corresponds to a result of the dyad [EMAIL PROTECTED])). Anyways, thanks for spelling this out. I'm going to have to review it a few times to make sure I've digested it all.
However, I believe you are using the words sample and sampling in a nonstandard sense. Here's what I think you are doing. You have a known population distribution f. You now take a vector v of length n in which x appears c(x) times, with the property that c(x)/n is approximately f(x). You regard this as a set of equiprobable outcomes, and so determines a population with distribution g satisfying g(x)=c(x)/n. This population has \mu=\sum x g(x)=(1/n)\sum x c(x) \sigma^2=\sum (x-\mu)^2 g(x)=(1/n)\sum (x-\mu)^2 c(x). So in this case using a denominator of 1/n makes sense. However, this is not the sample variance in any normally accepted sense: there is no sample in sight. There are n equiprobable outcomes defining a random variable X with P(X=x)=c(x)/n. Let me know if this is what you are getting it.
Yes, although I was not originally aware of the distinction between sample variance and population variance. (Or, at least, I had long since forgotten about that distinction.) Now that I have become acquainted with this distinction, I would classify this mechanism as describing population variance and not sample variance. I should probably also mention that the artifice you characterize as f(x) = c(x)/n is just one of several approaches which I have been considering -- I focused on it because it seemed computationally simple. Anyways, I think you've given me a decent set of tools for talking about populations, sampling and individual samples. Thanks, -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
