Momen (matematika)

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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

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

di mana X nyaéta variabel random nu mibanda sebaran ieu.

Momen tengah kan distribusi probabiliti variabel random X nyaéta

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

and

and

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.