# Approximating $Pr[n \leq X \leq m]$ for a discrete distribution

What's the best way to approximate $Pr[n \leq X \leq m]$ for two given integers $m,n$ when you know the mean $\mu$, variance $\sigma^2$, skewness $\gamma_1$ and excess kurtosis $\gamma_2$ of a discrete distribution $X$, and it is clear from the (non-zero) measures of shape $\gamma_1$ and $\gamma_2$ that a normal approximation is not appropriate?

Ordinarily, I would use a normal approximation with integer correction...

$Pr[(n - \text{½})\leq X \leq (m + \text{½})] = Pr[\frac{(n - \text{½})-\mu}{\sigma}\leq Z \leq \frac{(m + \text{½})-\mu}{\sigma}] = \Phi(\frac{(m + \text{½})-\mu}{\sigma}) - \Phi(\frac{(n - \text{½})-\mu}{\sigma})$

...if the skewness and excess kurtosis were (closer to) 0, but that's not the case here.

I have to perform multiple approximations for different discrete distributions with different values of $\gamma_1$ and $\gamma_2$. So I'm interested in finding out if there is an established a procedure that uses $\gamma_1$ and $\gamma_2$ to select a better approximation than the normal approximation.

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This is an interesting question, which doesn't really have a good solution. There a few different ways of tackling this problem.

1. Assume an underlying distribution and match moments - as suggested in the answers by @ivant and @onestop. One downside is that the multivariate generalisation may be unclear.

2. Saddlepoint approximations. In this paper:

Gillespie, C.S. and Renshaw, E. An improved saddlepoint approximation. Mathematical Biosciences, 2007.

We look at recovering a pdf/pmf when given only the first few moments. We found that this approach works when the skewness isn't too large.

3. Laguerre expansions:

Mustapha, H. and Dimitrakopoulosa, R. Generalized Laguerre expansions of multivariate probability densities with moments. Computers & Mathematics with Applications, 2010.

The results in this paper seem more promising, but I haven't coded them up.

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