In article <[EMAIL PROTECTED]>,
Richard A. Beldin <[EMAIL PROTECTED]> wrote:
>The suits and ranks of cards in a bridge deck certainly can be presented
>as independent sample spaces which we use as components of a cartesian
>product. Whether one does so or not is a matter of choice. I am on
>record as favoring the presentation as the cartesian product. Even the
>sample mean and variance can be seen this way, in fact, every vector
>valued random variable can be cast in the form of a random vector from a
>cartesian product.
This is the case for ONE card. Now suppose that one takes a
sample without replacement; it still is the case that the
suit of one card and the rank of another are independent, but
it is not the case that the number of cards of a given suit
and the number of cards of a given rank are independent.
>My point is that if we introduce independence as an attribute of sample
>spaces which we proceed to study as one, we can better motivate the idea
>of independent random variables and independent events.
How about this one, I believe due to Mandel? Take a sample
from a trivariate independent normal distribution. Then
each pair of correlations is independent, but the three
correlations cannot be.
Or this one, which leads to an easy derivation of the
Wishart distribution, and generation of Wishart matrices?
Let the sum of squares and cross products from a sample
of size n from a p-dimensional normal distribution with
mean 0 and covariance matrix I be written as AA', with
A 0 above the main diagonal. Then if n>=p, (the changes
for n<p are straightforward), the i-th diagonal element
of A has a chi distribution with n-i+1 degrees of freedom,
the subdiagonal elements are normal(0,1), and all of these
are independent.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
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