# Problem with Probabilistic

1. Aug 7, 2006

### eljose

I mean if you define the n-th momentum of a probability distribution:

$$<x^n > = \int_{-\infty}^{\infty} dx P(x) x^n$$ (1)

then my question is about the function P(x) (probability distribution) :surprised :surprised when:

$$P(x)=e^{-af(x) }$$ where a is a constant (real or complex)

-if f(x) is f(x)=x^2 then you have a "Gaussian" for a>0 and you can obtain every moment using (1)

- if f(x) is any function and a>0 ,a-->oo (big) you can use "Saddle-point approximation".

-the question is how can you handle if P(x) is of the form:

$$P(x)=x^2 + \alpha g(x)$$ where "alpha" is an small coupling constant so you could expand $$exp(\alpha g(x) )$$ and take only a few terms in that case i think you can take:

$$\int_{-\infty}^{\infty} dx P(x) = \sum_ n a(n) <x^n >$$