In article <[EMAIL PROTECTED]>,
Jerry Dallal <[EMAIL PROTECTED]> wrote:
>Herman Rubin wrote:
>> I also doubt
>> whether learning to compute answers gives any insight
>> into the concepts, except for those with good research
>> potential, and even there it tends to confuse.
>It depends on what "learning to compute" means. (*I'm* saying this
>in repsonse to a comment from Prf. Rubin?!)
>Consider exp(i pi).
>I can compute it by using Euler's rule or by viewing it as the pi
>radians rotation of a rod of unit length in the imaginary plane.
The second is not a means of computing, but of interpretation.
>Or consider the variance. I can compute it by using the desk
>calculator algorithm or by summing the squares of deviations.
Knowing how to do it, and why, is not the same as the actual
process of computing. I would go so far as to say that there
is little, if any, point about computing the SAMPLE mean and
the SAMPLE standard variance before understanding that of the
"population" mean. Even population here is a bad term, as it
implies that sampling without replacement is to be used, which
is not the same as that of numerical functions of observations
from arbitrary probability models. Even expectations should
be done on a "sample" space, and it should be shown, or at
least pointed out, that which equivalent formulation is used
leads to the same results, including using the distribution as
a particular one of these.
>If learning to compute means simply that one is given a formula--any
>formula--that is to be used without any thought of its origins, I
>agree.
>OTOH, thoughts about the method of computation can often lead to
>important insights.
It is SOMETIMES the case that the procedure, not the method
used to implement it, can do this. Setting up expectation
on sample spaces makes additivity trivial; pointing out the
equivalence of different representations makes expectation
and variance of the binomial and hypergeometric quite easy,
natural, and understandable.
However, doing it using combinatorics provides no insight
whatever, nor does using the cdf or pdf add much insight
to anything about the concepts.
Also, it is not necessary to introduce bivariate distributions
to develop covariance, or its properties. Expectations of
products are expectations, and simple algebra still works.
--
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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