From: jerry@hnrc.tufts.edu (Jerry Dallal)
Newsgroups: bit.listserv.edstat-l
Subject: Re: what's a random variable
Date: 3 Oct 92 19:34:18 GMT
Organization: USDA HNRC at Tufts University

In article <921002041726_71020.1025_EHC52-1@CompuServe.COM>, 71020.1025@COMPUSERVE.COM (William Kahn) writes:
> 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 . . .
> 
C'mon, Bill.  [For outsiders: I know Bill personally and *like* and *respect*
him a lot.]

One can ask that a proof be consistent . . . with something.  Which is why
there are frequentists and Bayesians and heavens knows what else.  If some 
respondents wish to adopt a Bayesian framework, fine.  (Boy, won't their
priors get updated quickly?)

But most respondents, for better or worse, have been trained as frequentists
and, therefore, must resolve the paradox in a frequentist framework.  In your
earliest post, you said these problems cause trouble because you often
condition on something without realizing it.  It this case, it was the amount
of money in the chosen envelope. 

> 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.

Okay, but in the envelope paradox you've looked in the envelope.  That matters.

> 
> 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?
> 
Good.  If you want t take a Bayesian approach, fine.  My point is that it's not
a frequentist approach.

> 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 $100 in Envelope A I change my
> probability.  You decide no longer to bet.  Is that kosher?  Do you not feel
> an obligation to assign betting odds to the contents of Envelope B?
> 
> What I'm not providing is a satisfactory calculus for manipulating my
> probabilities.  What you don't seem to provide (with a confidence interval
> or with the statement of p=1 or p=0 that B=200) is guidance (probabilities)
> where it matters.  (even as I write that down I scream out that p is
> "clearly" a random variable).
> 
> As a final wild lashing out at mad-dog frequentists of the world, how can
> confidence intervals possibly be taught in beginning statistics classes?
> They are so marginally useful, so abstract, so cleverly subtle, so easily
> misused (thankfully), that to routinely cover them virtually assures our
> profession of the continued ignoblity we have earned the past 40 years.
> 
Good.  Lash out at the frequentists!  Get in touch with the inner you.  Be the
best Bayesian you can be! :-)  We have no argument.  I say the solution to the
paradox for the frequentist is that E($)=.5*50$+.5*200$ is no more allowed than
"there is 95% probability the population mean lies in (36.7, 45.2)." There's MY
mathematics.  I've said, "Frequentists, you've made a mistake.  What random
variable gives rise to these probabilities?"

P.S.  It's easy to teach confidence intervals in an intro stat class.  
What follows is a few hours of lecture compressed into a few sentences.
A p% confidence interval is a member of a class of which p% achieve their
stated objective.
Consider (X1,X2) iid U(theta-1, theta+1).  Clearly (min,max) is a 50%
confidence interval.  Yet, if max-min>1 the interval *must* contain theta.

How  can you *not* teach CIs in an intro course?  After you teach someone how
to estimate something the next question is, "How accurate is it?"  How would
you go about answering?  CI's may not be perfect, but they are a widely used
method of answering the question.  Students should be familiar with them.
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