# Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$.

To find the $m$th central moment ("$m$th moment about the mean"), e.g. $m=2$ for the variance, you need to evaluate the MGF twice (once for $m=1$, once for $m=2$) and use the relationship:

$\text{var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$

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Is there an alternative kind of generating function such that to find the $r$th central moment ("$r$th moment about the mean") you only need to evaluate that generating function a single time?

• In passing, the Wikipedia page for cumulants indicates that the central moment generating function is given by $C(t)=\mathbb{E}[e^{t(x-\mu)}]=e^{-\mu t}M(t)$ where $\mu=\mathbb{E}[x]$ and $M(t)=\mathbb{E}[e^{t x}]$ is the regular MGF. So once you know $\mu$ and the usual MGF, the central MGF follows immediately. Commented Jun 17, 2016 at 19:58
• Actually that's exactly the kind of thing I was looking for. Commented Jun 17, 2016 at 21:55