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Estimating a Probability Distribution

  1. Aug 26, 2012 #1
    This is hopefully a simple question...

    Given the first n moments or central moments or cumulants (I don't care which) of a probability distribution, is there a standard procedure for estimating its functional form?

    For example, I know that given the mean and variance of a distribution, it's fairly standard to assume that it's Gaussian. Is there a more general method?
  2. jcsd
  3. Aug 26, 2012 #2


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    Homework Helper

    Hm. Well, if you know the first n moments, ##\mu_k##, you can construct a polynomial function

    $$\tilde{\varphi}_X(t) = 1+\sum_{k=1}^n \frac{\mu_k}{k!} (it)^k,$$

    where ##i## is the imaginary unit, which would be a polynomial approximation to the characteristic function ##\varphi_X(t)## of your distribution. The characteristic function of a random variable x is defined as

    $$\varphi_X(t) = \mathbb{E}[\exp(itx)];$$
    if you know ##\varphi_X(t)##, you can find the distribution by inverse fourier transform, i.e.,

    $$\rho_X(x) = \int_{-\infty}^\infty \frac{dt}{2\pi} e^{-itx}\varphi_X(t).$$

    In your case, you could numerically inverse fourier transform your polynomial approximation ##\tilde{\varphi}_X(t)##, which would give you a numerical estimate of the shape of the probability distribution. It won't tell you if it's a standard distribution like the normal distribution or something, but you'll know what it looks like.

    (Since you mentioned probability distribution, I am assuming that your random variables are continuous, rather than discrete.)

    The references in this part of the wikipedia page on characteristic functions may be useful.
  4. Aug 30, 2012 #3
    Thanks, that's exactly the kind of thing I was looking for!
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