Does a scaling and shift of first two moments change higher moments too? For a given random variable $X$ with mean $\mu_{\mathrm{old}}$ and standard deviation $\sigma_{\mathrm{old}}$ I would like to perform a transformation $g$ to obtain a new random variable $Y := g(X)$ which has a desired mean and standard deviation $\mu_{\mathrm{new}}$ and $\sigma_{\mathrm{new}}$ respectively without (if possible) changing other characteristics of the distribution (higher moments). A simple calculation shows that the transformation $g$ has the form
$$Y = g(X) := \frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}X-\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}$$
This fulfills the condition above since
$$E[Y] = \frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}E[X]-\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}=\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}}-\frac{\mu_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}=\mu_{\mathrm{new}}$$
and
$$\operatorname{Var}(Y) = \frac{\sigma^2_{\mathrm{new}}}{\sigma^2_{\mathrm{old}}}\operatorname{Var}(X)=\sigma^2_{\mathrm{new}}.$$
Does this normalization and scaling change other moments? I did some simulation in R and it seems no to be the case, at least for kurtosis and skewness. What about higher moments? Are there any reference for this?
 A: Skewness and kurtosis are not actually moments of the random variable itself (either raw or central). They're central moments divided by the corresponding power of $\sigma$, to make them unit-free. 
e.g. The third raw moment $\mu_3' = E(X^3)$ is affected by both changes of location and scale, the third central moment is $\mu_3=E[(X-\mu)^3]$ is not affected by changes of location (since that's removed when $\mu$ is subtracted) but is affected by change of scale. The skewness, by contrast, is $\mu_3/\sigma^3$; it's effectively the third moment of a standardized version of the original random variable.
Let $\mu_k$ be the $k$-th central moment of $X$ and let $Y=c(X-\mu_X)$. Then $E(Y^k)=E([c(X-\mu_X)]^k) = E(c^k[(X-\mu_X)]^k) = c^kE([(X-\mu_X)]^k)=c^k\mu_k$
So while $\mu_3$ is affected by change of scale, $\mu_3/\sigma^3$ (third-moment skewness) is not.
Similar comments apply to raw fourth moments, central fourth moments and kurtosis and the same ideas extend to higher order moments - raw moments are affected by both kinds of change, central moments by scale changes only, and standardized central moments by neither. 
