hypothetical measure of variability similar to entropy Variance has the following properties:


*

*$Var(cX)=c^2Var(X)$

*For independent variables $Var(X+Y)=Var(X)+Var(Y)$.


Range of a rv has the following properties:


*

*$Range(cX)=|c| Range(X)$

*For independent variables $Range(X+Y)=Range(X)+Range(Y)$.


Entropy has the following properties:


*

*$H(cX)=H(X)$

*For independent variables $H(X,Y)=H(X)+H(Y)$.


My main question: Does there exist a characteristic of a random variable such that:


*

*$New(cX)=New(X)$

*For independent variables $New(X+Y)=New(X)+New(Y)$.


?
(the only difference is "plus" instead of a vector random variable)
Maybe something can be said if the first property is replaced by $New(cX)=|c|^3New(X)$?
 A: A characteristic with the desired properties (almost)
Variance is the second central moment: $$Var(x)=E(X-\mu)^2.$$
So when trying to find a characteristic that satisfies conditions 1 and 2, a natural first guess could be the third central moment:$$\mu_3(X)=E(X-\mu)^3.$$
By expanding the cube, it can be verified (I recommend that you try it!) that $\mu_3$ has the following properties, which are close to what you were hoping for:


*

*Homogeneity: $\mu_3(cX)=c^3\mu_3(X)$

*Additivity: $\mu_3(X+Y)=\mu_3(X)+\mu_3(Y)$ when $X$ and $Y$ are independent.



Beyond the third moment
At this point you would perhaps guess that for the $k$:th central moment $$\mu_k(X)=E(X-\mu)^k$$ it holds that


*

*$\mu_k(cX)=c^k\mu_k(X)$

*$\mu_k(X+Y)=\mu_k(X)+\mu_k(Y)$ when $X$ and $Y$ are independent.


This is however not the case. There is however a characteristic $\kappa_k$, of which the variance and the third central moments are special cases, that has both the homogeneity and additivity properties. This $\kappa_k$ is called the $k$:th cumulant. It is derived from the logarithm of the moment-generating function $E(e^{Xt})$:
$$ \mbox{log}(E(e^{Xt}))=\sum_{k=1}^\infty \kappa_k\frac{t^n}{n!}. $$
As an example, the above formula can be used to see that
$$\kappa_2(X)=Var(X)\\\kappa_3(X)=\mu_3(X)\\\mbox{and}\\\kappa_4(X)=\mu_4(X)-3Var(X)^2.$$
The Wikipedia page for cumulants contains a lot more information about them.
