In article <[EMAIL PROTECTED]>,
Richard A. Beldin <[EMAIL PROTECTED]> wrote:
>... I have tried in vain to find natural
>examples of independent random variables in a smple space not
>constructed as a cartesian product.
An important example theoretically is the independence of the sample
mean and the sample variance of a data set consisting of points drawn
independently from a Gaussian distribution. Now, you might be able
to view this in terms of a Cartesian product, but it's not obvious
that that's a natural view.
>I think that introducing the word "independent" as a descriptor of
>sample spaces and then carrying it on to the events in the product space
>is much less likely to generate the confusion due to the common informal
>description "Independent events don't have anything to do with each
>other" and "Mutually exclusive events can't happen together."
I think this would be a bad idea. Events can be independent without
being constructed to be independent in this way.
As a definition, "Independent events don't have anything to do with
each other" is dangerous because it leads one to think that
independence is a property of events as physical phenomena. For
instance, one might decide that the event of a person having a
harmless variant of gene A is independent of the event of their having
a harmless variant of gene B, on the grounds that the mechanisms for
the two genes mutating are such that there's no reason for them to
mutate together. But if the genes are linked, and the context is a
sample of people from some community founded not too long ago by a
small number of people, the events of the two variants occuring in a
person may not be independent, even though they would be independent
if the context were a sample of people from the whole world. Here,
independence is not a property of the people, or of the genes, but of
what is considered to be the sample space for whatever problem is
being tackled.
Regarding "Mutually exclusive events can't happen together", this is
not an adequate definition if some non-null events have zero probability.
I think that independence is not something that can be explained in
ANY simple way. Multiple explanations and multiple examples are needed.
Radford Neal
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Radford M. Neal [EMAIL PROTECTED]
Dept. of Statistics and Dept. of Computer Science [EMAIL PROTECTED]
University of Toronto http://www.cs.utoronto.ca/~radford
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