Newsgroups: bit.listserv.edstat-l
Date:         Fri, 2 Oct 1992 00:17:26 EDT
Reply-To:     William Kahn <71020.1025@COMPUSERVE.COM>
From:         William Kahn <71020.1025@COMPUSERVE.COM>
Subject:      what's a random variable

Jerry Dallal writes:

>. . . <stuff deleted> . . .
>
>Give me mathematics
>
>. . . <stuff deleted> . . .
>
>Give me a RANDOM VARIABLE.
>
>. . . <stuff deleted> . . .

Jerry--Now I'm in a bind, not knowing what mathematics is nor what a random
variable is.  When I was in school I asked one of my favorite professors
(David Pollard for the record though all I say is strictly my version, his
is certainly different) what a mathematical proof was (I was having trouble
proving stuff).  i.e. how do you prove a proof is correct.  After all, now
and then published proofs turn out not to be incorrect, i.e. not proofs.
David said (my words) that a proof is whatever the editors of the current
mathematics journals said it was--i.e. an argument which convinced them ("I
can't define pornography but I know it when I see it" type of point). David
was an editor of some journals and thus what he said was a proof was a
proof. Shushed me right up and I went back to my sigma fields. Well, you
want mathematics, but it has been defined for me in such a way that, not
being an editor of a math journal, I can't know it when I see it. All I know
are arguments which change my mind. Which brings me to . . .

What IS a random variable?  Kolmogorov not withstanding, I consider the
outcome of a flipped but unobserved coin a random variable.  Does it really
make a difference if the coin is yet to be flipped?  What if the coin is in
the air and spinning?  Given its physical state it will hit the floor and
bounce in a deterministic, i.e. non-random, way.  I model lots of
deterministic phenomona as random variables all the time in industrial
problems.  Throw all those sums-of-squares into an "error" term and call it
"natural variability".  Pretty useful models.  Truth?  No way.  But useful.

The amount of money in Envelope B, is a random variable before I open
Envelope A.  The amount of money in Envelope B is again a (different) random
variable after I open Envelope A.  (By my definition of random variable).
Now, to be useful, I must develop a calculus for manipulating these animals
I choose to call random variables.  I say that the probability that Envelope
B has $200 (if Envelope A had $100) is 1/3.  Ah, but what is probability?
It is what I would bet at (given linear utility). I guess I got to go read
my deFinitti (subjective probability), huh?

What odds would you bet at that Envelope B has $200 after seeing Envelope A
has $100?  Before seeing Envelope A we would both bet at p=1/2 that Envelope
B has the larger amount.  After seeing the $100 in Envelope A I change my
probability.  You decide no longer to bet.  Is that kosher?  Do you not feel
an obligation to assign betting odds to the contents of Envelope B?

What I'm not providing is a satisfactory calculus for manipulating my
probabilities.  What you don't seem to provide (with a confidence interval
or with the statement of p=1 or p=0 that B=200) is guidance (probabilities)
where it matters.  (even as I write that down I scream out that p is
"clearly" a random variable).

As a final wild lashing out at mad-dog frequentists of the world, how can
confidence intervals possibly be taught in beginning statistics classes?
They are so marginally useful, so abstract, so cleverly subtle, so easily
misused (thankfully), that to routinely cover them virtually assures our
profession of the continued ignoblity we have earned the past 40 years.

Hey, and you thought you were excitable.  :-)

Bill Kahn
W. L. Gore and Associates
71020.1025@compuserve.com
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