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Motivated by some possible interpretation of some problem in physics, I have the following integral:

Let $X= X_1..X_n$ be a set of fixed real numbers.

The integral is : $I(X) = \int Dp\exp (- V_p(X)) $,

where $Dp = dp_1..dp_n \delta( \sum^n_{i=1}p_i-1) $, here the $ p_i$ are probabilities, real positive quantities in the interval $[0,1]$.

and $ V_p(X) $ is the variance of $ X$, relatively to the probability law $p$, that is: $ V_p(X) = \sum^n_{i=1}p_i X_i^2 - ( \sum^n_{i=1}p_iX_i )^2 $

I am interested, not mainly by the precise value of the quantity $ I(X)$, but more by its hypothetical meaning in some maths realm.

So, in maths, does $I(X)$ mean something, or/and is it useful for something , or/and is there something resembling ?

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  • $\begingroup$ the Dirac delta function $Dp$ with constraint that all probabilities sum to 1 makes me think the integral represents the gamut of various probabilities across the $n$ variables $\{ X \}$ $\endgroup$ – phdmba7of12 Nov 8 at 14:20
  • $\begingroup$ does $I$ represent a partition function of sorts .... $\endgroup$ – phdmba7of12 Nov 8 at 14:22
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    $\begingroup$ @phdmba7of12 The original context has nothing to do with probabilities...it is in some context of scattering amplitudes. $\endgroup$ – Trimok Nov 8 at 14:28
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    $\begingroup$ I think what you have here is the flat version of the Dirichlet distribution over $\vec{p}$ and you are evalulating $E[exp(-Var(\vec{p}))]$. Sorry this doesn't really answer your question of what it means... but perhaps some more googling would uncover something? $\endgroup$ – antkam Nov 8 at 15:30
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    $\begingroup$ This thread deals with the expected entropy $E[H(\vec{p})]$ for the flat Dirichlet distribution. It is pretty normal to study the entropy of a distribution, or the variance of a distribution, but this is the first time I have seen a question asked about the $e^{-Var}$ of a distribution. $\endgroup$ – antkam Nov 8 at 15:40

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