Functio momenta generans

Functio momenta generans $M_X:ℝ\to {ℝ}^{+},t\mapsto M_X(t),$ variabilis fortuiti $X$ sive distributionis eius exsistit, si pro omnibus $n\in \mathbb{N}$ valores medii exspectati $\mathrm{E}(X^n)$ exsistunt. Tum $M_X$ sic definitur:

$M_X(t)=\mathrm{E}({\mathrm{e}}^{tX}).$

$M_X$ ad momenta calculanda adhibetur, quod expansione functionis exponentialis in seriem demonstratur:

$M_X(t)=\mathrm{E}\left({\mathrm{e}}^{tX}\right)=E\left(\sum _{n=0}^{\mathrm{\infty }}\frac{(tX{)}^n}{n!}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{n!}\mathrm{E}(X^n)=\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{n!}{m}_n,$

ubi $m_n=\mathrm{E}(X^n)$ est momentum variabilis fortuiti $X$ ordine $n$.

Haec expansio dicit quomodo momentum $m_k$ omnis ordinis $k\in \mathbb{N}$ calculari potest:

$m_k=\mathrm{E}(X^k)=\frac{{\mathrm{d}}^k}{\mathrm{d}{t}^k}{M}_X(t){|}_{t=0}.$

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

Semper momenta centralia, ${\mu }_k=\mathrm{E}([X-\mathrm{E}(X){]}^k)$, etiam ex relationibus ad momenta non-centralia, $m_k=\mathrm{E}(X^k)$, calculari possunt.

• Momentum (statistica)
• Fornix (statistica)