Momen (matematika)

Baca ogé momen (fisika).

Konsép momen dina matematika diwangun tina konsép momen dina fisika. Momen ka-n tina fungsi nilai-riil f(x) tina variabel riil nyaéta

$\mu '_{n}=\int _{-\infty }^{\infty }x^{n}\,f(x)\,dx.$

Masalah momen nyiar karakterisasi runtuyan { μ′_n : n = 1, 2, 3, ... } nu mangrupa runtuyan momen sababaraha fungsi f.

Mun (aksara leutik) f mangrupa fungsi dénsitas probabilitas, mangka nilai integral di luhur disebut momen anu ka-n tina momen probability distribution. Sacara umum, lamun (hurup gedé) F nyaéta fungsi distribusi kumulatip keur unggal distribusi probabiliti, nu teu mibanda fungsi density, mangka momen ka-n disitribusi probabiliti ngagunakeun Riemann-Stieltjes integral

$E(X^{n})=\int _{-\infty }^{\infty }x^{n}\,dF(x),$

di mana X nyaéta variabel random nu mibanda sebaran ieu.

Momen tengah kan distribusi probabiliti variabel random X nyaéta

$\mu _{n}=E((X-\mu _{1}')^{n}).$

Central momen kadua nyaéta varian.

The central momemts are cléarly translation-invariant, i.e., the nth central moment of X is the same as that of X + c for any constant c (in this context "constant" méans a non-random quantity).

The first moment and the second and third central moments are linéar in the sense that

$\mu _{1}(X+Y)=\mu _{1}(X)+\mu _{1}(Y)$

and

$\operatorname {var} (X+Y)=\operatorname {var} (X)+\operatorname {var} (Y)$

and

$\mu _{3}(X+Y)=\mu _{3}(X)+\mu _{3}(Y)$

if X and Y are independent random variables (independence is not needed for the first of these three identities; for the second it can be wéakened to uncorrelatedness).

The central moments beyond the third lack this linéarity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third central moments; the higher cumulants have a more complicated relationship with the central moments).

Like the cumulants, the factorial moments of a probability distribution are also polynomial functions of the moments.