Donald Burrill wrote:
>
> On Tue, 2 Sep 2003, Stan Brown wrote in part:
>
> SB> I started to write that at first, but I don't think it's true.
> SB> The scores on the test may very well be continuous.
>
> DB> You were right the first time, Stan. It is true: scores on a test
> DB> are NOT continuous. Ever.
>
> > Sorry, I don't buy that. Think "partial credit". Scores on an esay
> > exam or an exam that involves working problems can be subdivided and
> > are not necessarily integers. They can be subdivided indefinitely --
> > in principle.
>
> And in principle this subdivision only gets you to the set of rational
> numbers, not the set of real numbers; and the rational numbers are
> isomorphic to the set of integers.
Look, folks - does it take a mathematician (of all people)
to point out that the distinction between "continuous" and "discrete"
has - in real statistics, as opposed to collections of counterexamples
or first courses in probability theory - little to do with the topology
of
the set underlying the distribution? It's a practical distinction based
on the behaviour of "most" of that distribution viewed at the scale of
the precision of measurement and/or practical significance. The
topological concepts merely form useful conceptual labels.
In other words, the distribution you get by convolving a
Bernoulli distribution with a N(0,1E-12) is discrete for any
practical purpose (though topologically continuous). Incomes,
given to the nearest cent, are continuous (but topologically discrete).
That's not what we tell the students in Probability 200 (though if we're
smart we qualify what we do tell them immediately) but it's how
we proceed in the real world.
(And, by the way, the rational numbers - as a topological or metric
space - are *not* isomorphic to the set of integers. They have the same
cardinality (so are isomorphic as a set), but any 1-1 map between them
is
highly discontinuous in the Q -> Z direction. In particular, the
sequence (0.3, 0.33, 0.333 ...) converges to 1/3. No sequence of
integers converges unless it is eventually constant.)
-Robert Dawson
.
.
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