# Estimating a Probability Distribution

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?

Mute
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.

Thanks, that's exactly the kind of thing I was looking for!