The requirement for the CLT to hold is that there should be a mean
and st deviation for the background distribution. This I checked in
Introduction to the Theory of Statistics by Mood, Graybill and Boes
For a Cauchy distribution we can not determine the standard
deviation and thus taking random sampling from this technically
infinite population does not obey the CLT. I have created my own
sampling to persuade me of this and it holds.
However once we have swapped from a theoretically infinite
population to a finite though large population then the mean and st.
Deviation are calculable so the CLT holds. So for sampling from a
specific sample of a Cauchy distribution the CLT holds.
Jean M. Russell
To: [EMAIL PROTECTED]
Date sent: Sun, 23 Sep 2001 18:29:32 -0400
From: Rich Ulrich <[EMAIL PROTECTED]>
Organization: University of Pittsburgh
Send reply to: [EMAIL PROTECTED]
Subject: Re: what type of distribution on this sampling
> On Fri, 21 Sep 2001 12:47:33 -0400, Joe Galenko
> <[EMAIL PROTECTED]> wrote:
>
> >
> > The mean of a random sample of size 81 from a population of size 1
> > billion is going to be Normally distributed regardless of the
> > distribution of the overall population (i.e., the 1 billion).
> > Oftentimes the magic number of 30 is used to say that the mean will
> > have a Normal distribution, although that is when we're drawing from
> > an infinitely large population. But for the purposes of determining
> > the distribution of a mean, 1 billion is effectively infinite. And
> > so, 81 is plenty.
> >
> > But note that this is for a _random_ sample, not just any kind of
> > sample.
> >
> But what if I have a population of numbers that is made up of
> 1 billion draws from a Cauchy distribution? No one has ever
> defined this for me, and I have never tested it, but if you *have* a
> sample in hand, then you *can* compute a standard deviation even
> though in theory the variance is "undefined" or infinite.
>
> I does seem to me that you could take your billion numbers, and
> standardize them to the mean of 78 and SD of <whatever> and
> it won't change the nature, that the numbers belong to a Cauchy - and
> repeated sampling will *not* give a standard deviation-of- the-mean
> that is smaller than the standard deviation of the draw.
>
> I'm afraid that I am not giving an answer here - I am raising a
> question that I don't know the answer to.
>
> But -- without being Cauchy -- it must be true that you can always
> imagine an outlier extreme enough that your means will not be normal,
> or be very useful compared to some other description.
>
> --
> Rich Ulrich, [EMAIL PROTECTED]
> http://www.pitt.edu/~wpilib/index.html
>
>
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Jean M. Russell M.A. M.Sc. [EMAIL PROTECTED]
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