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

\mu'_n=\int_{-\infty}^\infty x^n\,f(x)\,dx.

Masalah momen nyiar karakterisasi runtuyan { μ′n : n = 1, 2, 3, ... } nu mangrupakeun 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 gede) F nyaeta fungsi distribusi kumulatip keur unggal distribusi probabiliti, nu teu mibanda fungsi density, mangka momen ka-n disitribusi probabiliti migunakeun Riemann-Stieltjes integral

E(X^n)=\int_{-\infty}^\infty x^n\,dF(x),

dimana X nyaeta variabel random nu ngabogaan sebaran ieu.

Momen tengah kan distribusi probabiliti variabel random X nyaeta

\mu_n=E((X-\mu_1')^n).

Central momen kadua nyaeta varian.

The central momemts are clearly 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" means a non-random quantity).

The first moment and the second and third central moments are linear 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 weakened to uncorrelatedness).

The central moments beyond the third lack this linearity; 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.