Témbongkeun sumber pikeun 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

:<math>\mu'_n=\int_{-\infty}^\infty x^n\,f(x)\,dx.</math>

'''Masalah momen''' nyiar karakterisasi runtuyan { &mu;&prime;<sub>''n''</sub> : ''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 [[cumulative distribution function|fungsi distribusi kumulatip]] keur unggal distribusi probabiliti, nu teu mibanda fungsi density, mangka momen ka-''n'' disitribusi probabiliti  ngagunakeun [[Riemann-Stieltjes integral]]

:<math>E(X^n)=\int_{-\infty}^\infty x^n\,dF(x),</math>

di mana ''X'' nyaéta [[variabel random]] nu mibanda sebaran ieu.

[[moment about the mean|Momen tengah]] ka'''''n''''' distribusi probabiliti variabel random ''X'' nyaéta

:<math>\mu_n=E((X-\mu_1')^n).</math>

Central momen kadua nyaéta [[varian]].

The central momemts are cléarly translation-invariant, i.e., the ''n''th 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

:<math>\mu_1(X+Y)=\mu_1(X)+\mu_1(Y)</math>

and

:<math>\operatorname{var}(X+Y)=\operatorname{var}(X)+\operatorname{var}(Y)</math>

and

:<math>\mu_3(X+Y)=\mu_3(X)+\mu_3(Y)</math>

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

The central moments beyond the third lack this linéarity; in that respect they differ from the [[cumulant]]s (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 moment]]s of a probability distribution are also polynomial functions of the moments.

[[Kategori:Téori probabilitas]]
[[Kategori:Analisis matematik]]