لحظة مركزية

لكل n ، تكون العزم المركزي من الدرجة n متجانسة من الدرجة 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 n-th central moment, but continues to have this additivity property even when n ≥ 4 is the n-th cumulantκ_n(X). For n = 1, the n-th cumulant is just the expected value; for n = either 2 or 3, the n-th cumulant is just the n-th central moment; for n ≥ 4, the n-th cumulant is an n-th-degree monic polynomial in the first n moments (about zero), and is also a (simpler) n-th-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 n-th-order moment about the origin to the moment about the mean is

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

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

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

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 ${\displaystyle \operatorname {Var} (X)=\operatorname {E} [X^{2}]-\left(\operatorname {E} [X]\right)^{2}}$

$${\displaystyle {\begin{aligned}\mu _{3}&=\mu '_{3}-3\mu \mu '_{2}+2\mu ^{3}\\\mu _{4}&=\mu '_{4}-4\mu \mu '_{3}+6\mu ^{2}\mu '_{2}-3\mu ^{4}.\end{محاذاة}}}$$

... 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

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

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

في التوزيعات المتناظرة حول متوسطاتها (غير المتأثرة بانعكاسها حول المتوسط)، فإن جميع العزوم المركزية الفردية تساوي صفرًا كلما وجدت، لأنه في صيغة العزم النوني ، كل حد يتضمن قيمة X أقل من المتوسط ​​بمقدار معين يلغي تمامًا الحد الذي يتضمن قيمة X أكبر من المتوسط ​​بنفس المقدار.