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Problem with Probabilistic

  1. Aug 7, 2006 #1
    I mean if you define the n-th momentum of a probability distribution:

    [tex] <x^n > = \int_{-\infty}^{\infty} dx P(x) x^n [/tex] (1)

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

    [tex] P(x)=e^{-af(x) } [/tex] 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:

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

    [tex]\int_{-\infty}^{\infty} dx P(x) = \sum_ n a(n) <x^n > [/tex]
     
  2. jcsd
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