I have my moments

[Robert Israel]

I remember doing this, maybe about a year ago, and maybe posting it to
sci.math, but I can't seem to find it using Google Groups, so maybe I
didn't post it. Does anybody remember seeing it? Is my memory playing
tricks on me, or am I not searching the right keywords? The result is
well-known, I'm sure, but still it's cute:

Let X be the number of fixed points in a random permutation of n letters.
Then X and the Poisson distribution with parameter 1 have the same moments
up to and including the n'th.

Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[David Bernier]

Robert Israel wrote:
> I remember doing this, maybe about a year ago, and maybe posting it to
> sci.math, but I can't seem to find it using Google Groups, so maybe I
> didn't post it. Does anybody remember seeing it? Is my memory playing
> tricks on me, or am I not searching the right keywords? The result is
> well-known, I'm sure, but still it's cute:
>
> Let X be the number of fixed points in a random permutation of n letters.
> Then X and the Poisson distribution with parameter 1 have the same moments
> up to and including the n'th.

[...]

I can find something of the sort at mathforum.org for the
newsgroup sci.math.symbolic, but so far not for sci.math
even though the headers included sci.math, if I'm not
mistaken. The question posed was:
" What phenomenon has a standard deviation of EXACTLY 1.00? "
in the title, and the relevant post at mathforum seems to
be:
< http://mathforum.org/kb/message.jspa?messageID=7087030 > .

David Bernier

P.S. My first "hit" was through searching at mathKB:
< http://www.mathkb.com/ > on:
poisson israel
Page 4 returns the thread about standard deviation exactly 1.00 .

[David Bernier]

David Bernier wrote:
> Robert Israel wrote:
>> I remember doing this, maybe about a year ago, and maybe posting it to
>> sci.math, but I can't seem to find it using Google Groups, so maybe I
>> didn't post it. Does anybody remember seeing it? Is my memory playing
>> tricks on me, or am I not searching the right keywords? The result is
>> well-known, I'm sure, but still it's cute:
>>
>> Let X be the number of fixed points in a random permutation of n letters.
>> Then X and the Poisson distribution with parameter 1 have the same
>> moments
>> up to and including the n'th.
> [...]

There's a Wikipedia article on "Dobinski's formula",
which is:
(1/e) sum_{k = 0, ... infty} k^n/k! = B_n ,

the n'th Bell number.

Replacing n by a positive real number x gives:

(1/e) sum_{k= 0, ... infty} k^x/k! = a function of x > 0.

It seems this should give a quite "nice" interpolation formula
for the Bell numbers.

The article on Dobinski's formula can be found here:
< http://en.wikipedia.org/wiki/Dobinski's_formula > .

> I can find something of the sort at mathforum.org for the
> newsgroup sci.math.symbolic, but so far not for sci.math
> even though the headers included sci.math, if I'm not

I misread the headers. It wasn't in sci.math .

<< Newsgroups:
sci.math.symbolic,alt.sci.math.probability,alt.sci.math.statistics.prediction >> .

[Herman Rubin]

There are several ways to get this result. The easiest
is to compute the factorial moments, E(X!/(X-k)!), which
is k! times the expected number of sets of k coincidences.
But the number of sets of k potential coincidences is the
binomial coefficient C(n,k), and the probability of k a
particular set is (n-k)!/n!, as long as k <= n. So all
of these are 1.

On the other hand, the generating function of the Poisson(1)
distribution ie exp(t), and the k-th factorial moment is
the k-th derivative of the generating function at 0, which
is easily seen to be 1 for all k.

--
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
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

[Ray Koopman]

http://groups.google.ca/group/sci.math.symbolic/msg/3ab19e822e68267f

[Bill Taylor]

RI wrote:

> Does anybody remember seeing it?

I don't recall, but I wrote about this a great many years ago.

> Let X be the number of fixed points in a random permutation of n letters.
> Then X and the Poisson distribution with parameter 1 have the same moments
> up to and including the n'th.

Yes, it's a very cool result. I coined the phrase
"convergent in moments" for this type of convergence,
i.e. of permfix(n) to Poisson(1) as n --> oo.
_____________________________

Oddly enough, (synchronicity!), I stumbled on another cute
property of this distribution just yesterday! I expect it's well
known.

That is, a property of the Poisson(1) distribution.

Let X be chosen from the above distribution;
THEN let Y be uniformly chosen from the integers {0, 1, 2, ... ,X, X
+1 }.

What is the distribution of Y?

ANSWER:- It is exactly the same as for X! i.e. Poisson(1).

"""""""""""""""""""""""""""""

This cropped up as the answer to a very slowly increasing population
process.

Whatever the starting distribution (within reason), the population
tends
toward that same distribution, when the transition probs are as above.

KYUTE!

-- Bouncy Bill

** Postmodernists still fly in aeroplanes rather than on broomsticks,
** even though they "think all views are equally valid".