In article <[EMAIL PROTECTED]>,
Richard A. Beldin <[EMAIL PROTECTED]> wrote:
>You missed the point, Herman. I don't assert that these are independent
>random variables. I claim that introducing students to the concept of
>independent sample spaces from which we construct a cartesian product
>sample space will make it easier for them to understand independent
>events and random variables when we define them late.
I believe that this will not do what is expected, and might
even make it worse.
When we introduce sample spaces, we do not, and should not,
introduce the probabilities at that time. If we did, we
could not have inference, and also I believe that we need
to get across the idea that there is no "right" sample space
for a problem, but merely adequate representations; the
point in a sample space can represent the result of the
experiment under consideration, but we might have more.
Otherwise, how can we consider the number of successes to
be a real-valued random variable?
Sample spaces can be Cartesian products without the
coordinates being independent; whenever we have a bivariate
classification, we have a Cartesian product, whether or not
there is independence. We do not want students to consider
race and lactose intolerance to be independent.
Presenting oversimplified special cases seems to make it
harder for people to understand. I deliberately postpone
all considerations of symmetry or equally likely, as the
students (and also those using probability and statistics)
have a major tendency to impose this when it is very
definitely not the case. The "principle of insufficient
reason" contributed to the demise of Bayesian statistics
in the 19th century, and I see it going strong now.
--
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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