From: Kjetil Brinchmann Halvorsen <kjetil_at_acelerate.com>

Date: Thu 07 Jul 2005 - 12:43:46 EST

Date: Thu 07 Jul 2005 - 12:43:46 EST

Spencer Graves wrote:

>Hi, Göran: I'll bite:

*>
**> (a) I'd like to see your counterexample.
**>
**> (b) I'd like to know what is wrong with my the following, apparently
**>defective, proof that they can't be independent: First consider
**>indicator functions of independent events A, B, and C.
**>
**> P{(AC)&(BC)} = P{ABC} = PA*PB*PC.
**>
**> But P(AC)*P(BC) = PA*PB*(PC)^2. Thus, AC and BC can be independent
**>only if PC = 0 or 1, i.e., the indicator of C is constant almost surely.
**>
**> Is there a flaw in this?
**>
*

As far as I can see, this is correct.

> If not, is there some reason this case

*>cannot be extended the product of arbitrary random variables X, Y, and
**>W=1/Z?
**>
**>
*

Yes. Random variables are independent if all events which can be defined
in terms of them are independent.

If Z is non-constant, it must be some event defined by Z with
probability strictly beteween 0 and 1

and the above argument cannot be used.

Kjetil

> Thanks,

*> spencer graves
**>
**>Göran Broström wrote:
**>
**>
**>
**>>On Wed, Jul 06, 2005 at 10:06:45AM -0700, Thomas Lumley wrote:
**>>(...)
**>>
**>>
**>>
**>>>If X, Y, and Z are
**>>>independent and Z takes on more than one value then X/Z and Y/Z can't be
**>>>independent.
**>>>
**>>>
**>>Not really true. I can produce a counterexample on request (admittedly
**>>quite trivial though).
**>>
**>>Göran Broström
**>>
**>>______________________________________________
**>>R-help@stat.math.ethz.ch mailing list
**>>https://stat.ethz.ch/mailman/listinfo/r-help
**>>PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
**>>
**>>
**>
**>
**>
*

-- Kjetil Halvorsen. Peace is the most effective weapon of mass construction. -- Mahdi Elmandjra -- No virus found in this outgoing message. Checked by AVG Anti-Virus. ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Thu Jul 07 13:23:53 2005

*
This archive was generated by hypermail 2.1.8
: Fri 03 Mar 2006 - 03:33:18 EST
*