This repo implements formulas for high-order moments of binomial distribution, discussed in the paper "Handy formulas for Binomial Moments".
Here is how the sixth moment looks like, as a function of the probability parameter and the number of trials.
And here are some first central moments, with $q=1-p,\sigma^2=pq$:
| $d$ |
$\mathbb{E}[(S-\mathbb{E}[S])^d],\quad S\sim\mathrm{Binom}(n,p)$ |
| 2 |
$n \sigma^{2}$ |
| 3 |
$n \sigma^{2} \left(- 2 p + 1\right)$ |
| 4 |
$3 n^{2} \sigma^{4} + n \left(- 6 \sigma^{4} + \sigma^{2}\right)$ |
| 5 |
$\left(- 2 p + 1\right) \left(10 n^{2} \sigma^{4} + n \left(- 12 \sigma^{4} + \sigma^{2}\right)\right)$ |
| 6 |
$15 n^{3} \sigma^{6} + n^{2} \left(- 130 \sigma^{6} + 25 \sigma^{4}\right) + n \left(120 \sigma^{6} - 30 \sigma^{4} + \sigma^{2}\right)$ |
| 7 |
$\left(- 2 p + 1\right) \left(105 n^{3} \sigma^{6} + n^{2} \left(- 462 \sigma^{6} + 56 \sigma^{4}\right) + n \left(360 \sigma^{6} - 60 \sigma^{4} + \sigma^{2}\right)\right)$ |
| 8 |
$105 n^{4} \sigma^{8} + n^{3} \left(- 2380 \sigma^{8} + 490 \sigma^{6}\right) + n^{2} \left(7308 \sigma^{8} - 2156 \sigma^{6} + 119 \sigma^{4}\right) + n \left(- 5040 \sigma^{8} + 1680 \sigma^{6} - 126 \sigma^{4} + \sigma^{2}\right)$ |
| 9 |
$\left(- 2 p + 1\right) \left(1260 n^{4} \sigma^{8} + n^{3} \left(- 13216 \sigma^{8} + 1918 \sigma^{6}\right) + n^{2} \left(32112 \sigma^{8} - 6948 \sigma^{6} + 246 \sigma^{4}\right) + n \left(- 20160 \sigma^{8} + 5040 \sigma^{6} - 252 \sigma^{4} + \sigma^{2}\right)\right)$ |
| 10 |
$945 n^{5} \sigma^{10} + n^{4} \left(- 44100 \sigma^{10} + 9450 \sigma^{8}\right) + n^{3} \left(303660 \sigma^{10} - 99120 \sigma^{8} + 6825 \sigma^{6}\right) + n^{2} \left(- 623376 \sigma^{10} + 240840 \sigma^{8} - 24438 \sigma^{6} + 501 \sigma^{4}\right) + n \left(362880 \sigma^{10} - 151200 \sigma^{8} + 17640 \sigma^{6} - 510 \sigma^{4} + \sigma^{2}\right)$ |
These formulas can be used to obtain skewness and kurtosis:
|
|
| skewness |
$\frac{1-2p}{\sqrt{npq}}$ |
| excess kurtosis |
$\frac{1-6pq}{npq}$ |
