In article <a02179a3-3235-4227-86d1-c221bf393... at (no spam) y21g2000hsf.googlegroups.com>,
pamela fluente <pamelaflue... at (no spam) libero.it> wrote:
On 7 Mag, 19:15, hru... at (no spam) odds.stat.purdue.edu (Herman Rubin) wrote:
In article <7532ba3b-6895-4d3b-9caa-e03009581... at (no spam) m36g2000hse.googlegroups.> >com>,
pamela fluente =A0<pamelaflue... at (no spam) libero.it> wrote:
for the distribution of X1 + ... +XN with Xi negative exponential
(with arbitrary parameters)
is there an exact formula for the =A0probability mass function and cdf ?
Can you please provide it or point me to a site or paper with this
exact formula (if it exists) ?
I can provide you with a formula; how useful it is may not
be all that clear. =A0I am presuming the X's are independent,
and the density of Xj is aj*exp(-aj*u) on 0, infinity.
The moment generating function of the sum is
1/prod(1 - t/aj), and in all cases, this has a
partial fraction decomposition, giving the
density and cumulative distribution function
as linear functions of the above functions,
multiplied by the usual polynmials for multiple
a's. =A0If the a's are all distinct, the j-th
density above should be multiplied by
1/prod'(1-aj/ak), where the product is taken]
over the other a's.
Thanks a lot, Herman!
did you just invent it on the fly or is there a book or a paper
I can correctly Reference, stating for the first time the exact
probability mass function
and cdf of the convolution of N negative exponential *with arbitrary
parameters* ?
If possible, I would really like a precise pointer or reference I
can verify and quote with exact attribution
of the result to the first Author who pointed out the above exact
expressions (pmf and cdf).
[In other words, who is the Author of the result, or the one who
should be correctly mentioned
in a research paper as the right dad of the results]
Thank you,
-P
Since the distribution is on the positive reals, and convolution
of two distributions is the product of the mgfs or Laplace
transforms, this could have been done at any time after that.
The quick computation of the coefficients comes from Lagrange
interpolation formula, equally old.
Whether anyone put these two well-known facts together for
an arbitrary convolution of exponentials with different
coefficients I do not know; from the above paragraph, you
may infer that I considered it "obvious". But this is
most1 of mathematical research; seeing the obvious, and
showing why it is.
--
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... at (no spam) stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558- Nascondi testo tra virgolette -
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