Richard A. Beldin wrote:
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> I have long thought that the usual textbook discussion of independence
> is misleading. In the first place, the most common situation where we
> encounter independent random variables is with a cartesian product of
> two independent sample spaces. Example: I toss a die and a coin. I have
> reasonable assumptions about the distributions of events in either case
> and I wish to discuss joint events. I have tried in vain to find natural
> examples of independent random variables in a sample space not
> constructed as a cartesian product.
>
> 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."
>
> Comments?
1) It is conceivable, that a plant making blue and red 'thingies' on
the same production line would discover that the probability that the
next thingie is internally flawed (in the cast portion) is independent
of the probability that it is blue.
BTW - 'Thingies' are so commonly used by everyone that it is not
necessary to describe them in detail. :)
2) There are many terms, concepts, and definitions in the 'textbook'
that have no exact match in reality. Common expressions include, "There
is no such thing as random,' 'There is no such thing as Normal
(distribution),' and my own contribution, "There is no such thing as a
dichotomy this side of a theological discussion.' The abstract
definitions are just that - theoretical ideals. Down here in the mud of
reality, we recognize this, and try to decide if the theory is
reasonably close to what is happening. A couple confirmation trials
help, too.
If the internal casting flaws are generated at an early point, and the
paint is added later, depending on the orders received, then I would
assert that independence was likely. If the paint is added to castings
made on different dies or production machines, as a color code, then I
would suspect independence was unlikely.
3) Presenting 'independence' as axes in a cartesian coordinate system
is extremely handy, especially for discussing orthogonal arrays and
designed experiments, etc. The presentation, however, does not make
them independent. One has to check the physical system behavior to
assure that.
4) I may have shot far wider than your intended mark, in which case,
sorry for the interruption.
Jay
--
Jay Warner
Principal Scientist
Warner Consulting, Inc.
4444 North Green Bay Road
Racine, WI 53404-1216
USA
Ph: (262) 634-9100
FAX: (262) 681-1133
email: [EMAIL PROTECTED]
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