In article
<1adc9e68-86fd-4ac4-ba5e-b375955ac8ce@[EMAIL PROTECTED]
>,
<mikko.kauppila@[EMAIL PROTECTED]
> wrote:
> Hello,
>For discrete r.v.s we have the probability function, e.g., P(X = 5).
>For continuous r.v.s. we have the density function, e.g., f(x).
>But is there any way to treat discrete probability functions
>and continuous density functions in a unified fa****on?
>That is, is there any "common word" for discrete probabilities
>and continuous densities?
Probability measures includes both of these and more.
The ideas, and even many of the results, of general
measure theory belong in high school, but not as
presently taught.
>Obviously, such treatment would be massively helpful
>when we don't wish to make assumptions
>about the continuity of RVs.
Agreed. The way that expectation is taught is
even worse. As I tell my students, everything
is discrete or limits of discrete.
>So far I have got two ideas, but I'm not sure
>whether they are statistically "meaningful"...
>1.) Use the term "density function" for both discrete
>and continuous r.v.s. There is some intuitive appeal
>behind this idea, but rigorous treatment is probably
>complicated, I would think.
If one has two "appropriate" measures f and g, and f is
absolutely continuous with respect to g, then for any
set A, f(A) is the integral of df/dg, the Radon-Nikodym
derivative of f with respect to g, over A.
>2.) Use the term "likelihood function" for both discrete
>and continuous r.v.s. Technically, a likelihood function
>is defined as the density (or probability) of given
>data with respect to an unknown parameter.
Not the density with respect to a parameter, but
the density with respect to a fixed distribution
given the parameter; see below.
>Therefore, this "trick" would neatly unify the
>treatment of continuous and discrete r.v.s,
>but at the cost of technical correctness, I would think.
If one has a family of distributions all absolutely
continuous with respect to the same appropriate
distribution, their density function with respect
to that distribution is a likelihood function. One
cannot do more, as it is only the ratio of likelihood
functions for different values of the parameter which
matter.
>I apologize if this sounds confusing, I wasn't originally
>trained as a statistician. Any help or literature
>pointers are greatly appreciated.
I suggest you read a text on measure theory; it should
all become clear.
In my opinion, probability should be taught in this
manner, even at the lowest levels, of course leaving
out many of the proofs at low levels. The understanding
of limits is im****tant, and belongs in elementary school
with the teaching of infinitely long decimal expansions.
Concentrate on the ideas; details can be looked up.
--
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, Department of Statistics, Purdue University
hrubin@[EMAIL PROTECTED]
Phone: (765)494-6054 FAX: (765)494-0558
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