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 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
dimana X nyaeta variabel random nu ngabogaan sebaran ieu.
Momen tengah kan distribusi probabiliti variabel random X nyaeta
- μn = E((X - μ1')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
- μ1(X + Y) = μ1(X) + μ1(Y)
and
and
- μ3(X + Y) = μ3(X) + μ3(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.