Fontem paginae Functio momenta generans inspicere

'''Functio momenta generans''' <math>M_X\colon\R\to\R^+,\ t\mapsto M_X(t),</math> [[variabile fortuitum|variabilis fortuiti]] <math>\,X</math> sive distributionis eius exsistit, si pro omnibus <math>n\in\N</math> [[valor medius exspectatus|valores medii exspectati]] <math>\operatorname{E}(X^n)</math> exsistunt. Tum <math>\,M_X</math> sic definitur:

:<math>M_X(t)=\operatorname{E}(\operatorname{e}^{tX}).</math>

<math>\,M_X</math> ad [[momentum (statistica)|momenta]] calculanda adhibetur, quod expansione functionis exponentialis in seriem demonstratur:

:<math>\,M_X(t)=\operatorname{E}\left(\operatorname{e}^{tX}\right)
             =E\left(\sum_{n=0}^{\infty}\frac{(tX)^n}{n!}\right)
             =\sum_{n=0}^{\infty}\frac{t^n}{n!}\operatorname{E}(X^n)
             =\sum_{n=0}^{\infty}\frac{t^n}{n!}m_n,</math>

:ubi <math>m_n=\operatorname{E}(X^n)</math> est momentum [[variabile fortuitum|variabilis fortuiti]] <math>\,X</math> ordine <math>\,n</math>.

Haec expansio dicit quomodo [[momentum (statistica)|momentum]] <math>\,m_k</math> omnis ordinis <math>k\in\N</math> calculari potest:

:<math>m_k = \operatorname{E}(X^k) = \frac{\operatorname{d}^k}{\operatorname{d} t^k} M_X(t)\biggr\vert_{t=0}.</math>

Si <math>\operatorname{E}(X)</math> parametro <math>\,\mu</math> familiae distributionum correspondet, momenta centralia eis momentis non-centralibus correspondent, quae ad parametrum <math>\,\mu=0</math> pertinent.

Semper momenta centralia, <math>\mu_k = \operatorname{E}([X-\operatorname{E}(X)]^k)</math>, etiam ex relationibus ad momenta non-centralia, <math>m_k = \operatorname{E}(X^k)</math>, calculari possunt.

{{NexInt}}
* [[Momentum (statistica)]]
* [[Fornix (statistica)]]

[[Categoria:Res statisticae]]