Central moment

The nth central moment is translation-invariant, i.e. for any random variable X and any constant c, we have

${\displaystyle \mu _{n}(X+c)=\mu _{n}(X).\,}$

For all n, the nth central moment is homogeneous of degree n:

${\displaystyle \mu _{n}(cX)=c^{n}\mu _{n}(X).\,}$

Only for n such that n equals 1, 2, or 3 do we have an additivity property for random variables X and Y that are independent:

${\displaystyle \mu _{n}(X+Y)=\mu _{n}(X)+\mu _{n}(Y)\,}$ provided n ∈ {1, 2, 3}.

A related functional that shares the translation-invariance and homogeneity properties with the nth central moment, but continues to have this additivity property even when n ≥ 4 is the nth cumulant κ_n(X). For n = 1, the nth cumulant is just the expected value; for n = either 2 or 3, the nth cumulant is just the nth central moment; for n ≥ 4, the nth cumulant is an nth-degree monic polynomial in the first n moments (about zero), and is also a (simpler) nth-degree polynomial in the first n central moments.

Sometimes it is convenient to convert moments about the origin to moments about the mean. The general equation for converting the nth-order moment about the origin to the moment about the mean is

${\displaystyle \mu _{n}=\operatorname {E} \left[\left(X-\operatorname {E} [X]\right)^{n}\right]=\sum _{j=0}^{n}{n \choose j}(-1)^{n-j}\mu '_{j}\mu ^{n-j},}$

where μ is the mean of the distribution, and the moment about the origin is given by

${\displaystyle \mu '_{m}=\int _{-\infty }^{+\infty }x^{m}f(x)\,dx=\operatorname {E} [X^{m}]=\sum _{j=0}^{m}{m \choose j}\mu _{j}\mu ^{m-j}.}$

For the cases n = 2, 3, 4 — which are of most interest because of the relations to variance, skewness, and kurtosis, respectively — this formula becomes (noting that ${\displaystyle \mu =\mu '_{1}}$ and ${\displaystyle \mu '_{0}=1}$):

${\displaystyle \mu _{2}=\mu '_{2}-\mu ^{2}\,}$ which is commonly referred to as ${\displaystyle \operatorname {Var} (X)=\operatorname {E} [X^{2}]-\left(\operatorname {E} [X]\right)^{2}}$

${\displaystyle \mu _{3}=\mu '_{3}-3\mu \mu '_{2}+2\mu ^{3}\,}$

${\displaystyle \mu _{4}=\mu '_{4}-4\mu \mu '_{3}+6\mu ^{2}\mu '_{2}-3\mu ^{4}.\,}$

... and so on,[2] following Pascal's triangle, i.e.

${\displaystyle \mu _{5}=\mu '_{5}-5\mu \mu '_{4}+10\mu ^{2}\mu '_{3}-10\mu ^{3}\mu '_{2}+4\mu ^{5}.\,}$

because ${\displaystyle 5\mu ^{4}\mu '_{1}-\mu ^{5}\mu '_{0}=5\mu ^{4}\mu -\mu ^{5}=5\mu ^{5}-\mu ^{5}=4\mu ^{5}}$

The following sum is a stochastic variable having a compound distribution

${\displaystyle W=\sum _{i=1}^{M}Y_{i},}$

where the ${\displaystyle Y_{i}}$ are mutually independent random variables sharing the same common distribution and ${\displaystyle M}$ a random integer variable independent of the ${\displaystyle Y_{k}}$ with its own distribution. The moments of ${\displaystyle W}$ are obtained as

${\displaystyle \operatorname {E} [W^{n}]=\sum _{i=0}^{n}\operatorname {E} \left[{M \choose i}\right]\sum _{j=0}^{i}{i \choose j}(-1)^{i-j}\operatorname {E} \left[\left(\sum _{k=1}^{j}Y_{k}\right)^{n}\right],}$

where ${\displaystyle \operatorname {E} \left[\left(\sum _{k=1}^{j}Y_{k}\right)^{n}\right]}$ is defined as zero for ${\displaystyle j=0}$.

In distributions that are symmetric about their means (unaffected by being reflected about the mean), all odd central moments equal zero whenever they exist, because in the formula for the nth moment, each term involving a value of X less than the mean by a certain amount exactly cancels out the term involving a value of X greater than the mean by the same amount.