Hi,
i need a general analytic and detailed PDF (probability density
function) of the randon variable Y, with:
Y = sum_{i=1}^{N}{X_i} + 1/sum_{k=1}^{M}{1/Z_k}
with X_i, Z_k iid random var. with exponentiel density. N, M positive integer.
Thanks. |
Clarification of Question by
torrent-ga
on
28 May 2005 09:32 PDT
Hi,
i make a mistake in my question. Indeed, instead of the above
formulation, i need the PDF of Y, with:
Y = sum_{i=1}^{N}{1/sum_{k=1}^{M_i}{1/Z_{i,k}}
with Z_{i,k} independent exponentielle random variables for each 'i'
not equal to 'k'.
Tanks.
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Clarification of Question by
torrent-ga
on
28 May 2005 09:35 PDT
Another correction:
------------------
Z_{i,k} independent exponentielle densities for all 'i' and 'k'.
thanks.
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Request for Question Clarification by
hedgie-ga
on
31 May 2005 05:35 PDT
torrent-ga
Let's see if I understand the question.
It is not easy to express formulas in ASCII.
1) Z_{i,k} : For i!=k these are independent random variables with
pdf of the type
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm
a) yes?
b) beta and mi are given for each i and k?
c) what if i==k ?
2) Y is sum of [ 1/S (k) ]
sum is over i =1,N exept i==k
where S(i) = sum over k { 1/ Z(i,k) }
sum is over k=1, 2, m(i)
b) m(i) is given
3) How big is N
a) How accurate it needs to be?
b) analytic and detailed
means what? By detailed you mean 'exact '?
If analytic expression does not exist - then what?
Hedgie
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Clarification of Question by
torrent-ga
on
31 May 2005 06:39 PDT
Hi,
Thank you for your help.
1) Z_{i,k} is an exponentielle density for all i = 1 to N, and all k = 1 to m(i).
i take exponentielle density equal to: 1/beta(i,k)*exp(-x/beta(i,k))
(which means that the mu == 0).
beta is given for each i and k.
i and k are independent parameters, so if i==k means nothing, only
that Z_{1,1} is an exponentielle density independent of other Z_{i,k}.
2)NO. Y is sum of 1/S(i) over i = 1 to N.
with S(i) = sum over k { 1/ Z(i,k) }; k = 1 to m(i)
N and m(i) are given.
3) How Big is N..Hmmm..not a big value but not a small one.
N and m(i) limited for example by 10 and 20 respectively..
However, i need an analytic expression of it, as small as possible.
If it didn't exist, it means there is no answer :).
I can simplify the problem for you.
One major point of the problem is to find the Inverse Laplace
Transform of a product of a modified second kind bessel functions,
like this:
F(p) = p^{M/2}*K_1(alpha1*sqrt(p))*K_1(alpha2*sqrt(p))*K_1(alpha3*sqrt(p))*...*K_1(alphaM*sqrt(p))
The Bessel function sqrt(x)*K_1(alpha1*sqrt(x)) is the Moment
Generator Function (MGF) of 1/X with X expoenentiel density.
I need a simplified analytic formulation, even if there is two or three integrals.
Thanks you and good luck.
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Clarification of Question by
torrent-ga
on
31 May 2005 18:58 PDT
Hi,
i read your link. Indeed, it's an Exponential Distribution. I write it
in French :), sorry.
For distribution the central limit is true for high values of N, and
not for every value of it. For N =1, 2, 3 and even until 20, it's not
a Gaussian distribution.
I didn't offer 30$ to take the central limit theorem :).
i Hope a more analytic solution, as i present to you in my last
clarification (Inverse Laplace Transform of a Bessel product).
In case if it's difficult for you, can you please let the question
open for your collegues, like Mr MathTalk-ga.
Thanks.
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