Hi MathTalk,
i have an integral equation problem which i resume in the following PDF file:
http://0201.netclime.net/1_5/X/H/G/Exercise_ANIBA_En.pdf
I hope that's your answer will be excellent as exepected :).

Thanks.

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Request for Question Clarification by mathtalk-ga on 20 Oct 2004 04:10 PDT

Hi, ghassane-ga:

In the case you have solved, for the exponential distributions, the
density functions are monotone decreasing from 0 to +oo.  I'm
wondering if the application you have in mind supports a similar
assumption.

Is there any additional information about the density functions or
cumulative distribution functions based on them which can be assumed?

regards, mathtalk-ga

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Clarification of Question by ghassane-ga on 20 Oct 2004 06:27 PDT

Hi,
No, the application didn't support similar assumption. In general,
i'll apply it for Gaussian, Exponential, Rayleigh, Nakagami,
log-normal distrinutions.
Thanks.

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Clarification of Question by ghassane-ga on 20 Oct 2004 06:37 PDT

Hi,
i supposes that you already know that cumulative distribution function
is an increasing function which converges in Infinity to 1.

ragrds, ghassane-ga

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Request for Question Clarification by mathtalk-ga on 20 Oct 2004 08:41 PDT

Yes, I'm thinking about a "change of variable" approach that would
simplify some of the expressions and establish some broad conditions
under which a solution r is known to exist (and perhaps be unique).

regards, mathtalk-ga

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Clarification of Question by ghassane-ga on 20 Oct 2004 13:55 PDT

Hi,
I'm sure that there existes a unique solution.
Please follow this link, to see the original form of the equation,
perhaps it will be simpler to found "r" than in the actual form.
http://0201.netclime.net/1_5/G/T/Q/Original_Function.pdf

regards, ghassane-ga.

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Clarification of Question by ghassane-ga on 20 Oct 2004 13:58 PDT

Hi,
can you add the Rician distribution to the other, if you found
solution for the above-mentionned distributions.
thanks.

regards, ghassane-ga.

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Clarification of Question by ghassane-ga on 22 Oct 2004 06:17 PDT

Hi mathtalk,
You can suppose that we have a finite expection E[X] and E[Y].
I know that you have already solved a problem like this one, and
that's why i choose you :).
nice weekend with my problem.

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Clarification of Question by ghassane-ga on 27 Oct 2004 08:43 PDT

Hi MathTalk,
are you still workin in my problem or you forgot it :))
thanks.

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Request for Question Clarification by mathtalk-ga on 27 Oct 2004 13:47 PDT

Well, I've sorted out the equivalence of the three ways you stated the
problem, and I came up with a fourth way of stating it myself that
makes existence/uniqueness more obvious.

As with the other Question you posted, it would probably be a good
time to clarify exactly what is expected in the nature of an Answer. 
You've given the closed form solution for the exponential
distributions f_X(x) and f_Y(y).  I think it unlikely that for the
list price offered I'd have time to do a proper investigation of all
five other distributions that you named:

Gaussian, Rayleigh, Nakagami, log-normal, Rice

if the intention is to find "closed form" solutions for each.

However if that's what you are hoping to obtain, I'd be agreeable to
open the Question up for other Researchers so that everyone has an
opportunity to try.

regards, mathtalk-ga

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Clarification of Question by ghassane-ga on 27 Oct 2004 15:05 PDT

Hi,
i agree with you. but, how can i open it to other researchers?

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Request for Question Clarification by mathtalk-ga on 27 Oct 2004 15:40 PDT

You just did!

best wishes, mathtalk-ga

I took a look at the "original form" and quite agree that from it, it
is certain that a unique solution exists under fairly broad
conditions.

The broad conditions, in the present circumstances, are that both
distributions f_X(x) and f_Y(y) have finite expectations E(X) and E(Y)
respectively.

The proof of unique solution can be generalized beyond the present
setting to a suitable joint distribution on x and y, using instead an
assumption of finite expectations on the two marginal distributions.

When I first read this problem, I thought there was perhaps a
superficial resemblance to an earlier Google Answers problem I solved:

[Integral Equation]
http://answers.google.com/answers/threadview?id=374299

Although the details were certainly different, I'm starting to suspect
that something of the approach taken there will be useful here.  Over
the weekend I hope to sort out the facts.  In the meantime I'll also
take a look at the other problem you posted.

regards, mathtalk-ga

Researchers and Commenters:

Please note that with my inability to provide a timely investigation
of all the issues raised by this Question, ghassane-ga has graciously
agreed to open the problem up to all interested parties.

regards, mathtalk-ga