# Meaning of a particular integral on the space of probabilities

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 ?

• 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 $\{ X \}$ – phdmba7of12 Nov 8 at 14:20
• does $I$ represent a partition function of sorts .... – phdmba7of12 Nov 8 at 14:22
• @phdmba7of12 The original context has nothing to do with probabilities...it is in some context of scattering amplitudes. – Trimok Nov 8 at 14:28
• 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? – antkam Nov 8 at 15:30
• 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. – antkam Nov 8 at 15:40