How to calculate the central moment giving function of a distribution Is there a function which gives the central moments instead of just moments of a distribution and if so how to calculate this function for a distribution e.g. the normal distribution. 
 A: The "function" follows directly from the definition of central moments:
$\qquad \mu_k=E[(X-\mu)^k]$, 
and the usual expectation of a function of a random variable $E[g(X)]=\int_{-\infty}^\infty g(x) f_X(x) dx$ (equivalently, see the law of the unconscious statistician), giving
$\qquad \mu_k=E[(X-\mu)^k]=\int_{-\infty}^\infty (x-\mu)^k f_X(x) dx$
This is just the definition given at the relevant Wikipedia page.
If one tries that for the normal case, $N(\mu,\sigma^2)$, a simple bit of manipulation reduces it to: $\mu_k = \sigma^k \int_{-\infty}^\infty x^k \phi(x) dx$ -- i.e. apart from a scaling factor of $\sigma^k$, this is just the raw $k$-th moment of the standard normal.
You can use symmetry (plus the finiteness of the positive half) to conclude that all the odd moments in the standard normal are 0. With even moments, you again use symmetry to write it as double the right half of the integral, and then you can use substitution (let $u=x^2$) to reduce that to a constant times a gamma integral.
