Absolute third central moment for standard distributions reference I have to write an R function that computes the absolute third central moment (i.e. $\mathbb{E}[|X-\mathbb{E}[X]|^3]$) in the cases that you are given the name of the distribution or the PMF/CDF. I know how to do this, but what I want is a reference for the absolute third central moment for distributions such as Bernoulli, Binomial, Geometric, Exponential, Gamma etc. (the pretty standard ones) so that I can easily check the output of my function. I don't know of any predefined R functions that do this either.

Here's a few values to get started with since it might occasionally be handy to have some third absolute moments to refer to. I've made this answer community wiki -- anyone should feel free to edit to include more if they wish. I will try to add more when I get some time.
In each case $\sigma^2$ is the variance of the distribution in question (this way the specifics of the parameterization doesn't matter).
Normal: $\sqrt{\frac{\pi}{8}}\,\sigma^3$ $\quad$  ($\approx 1.596$ for standard normal)
Exponential: $(\frac{12}{e}-2)\,\sigma^3$ $\quad$ ($\approx 2.415$ for standard exponential)
Uniform (continuous): $\frac{12^{3/2}}{32}\,\sigma^3$ $\quad$ ($\frac{1}{32}=0.03125$ for standard uniform)
(computed algebraically, checked by large simulation)
