Functio momenta generans

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

$M_{X}(t)=\operatorname {E} (\operatorname {e} ^{tX}).$

$\,M_{X}$ ad momenta calculanda adhibetur, quod expansione functionis exponentialis in seriem demonstratur:

$\,M_{X}(t)=\operatorname {E} \left(\operatorname {e} ^{tX}\right)=E\left(\sum _{n=0}^{\infty }{\frac {(tX)^{n}}{n!}}\right)=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}\operatorname {E} (X^{n})=\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}m_{n},$

ubi $m_{n}=\operatorname {E} (X^{n})$ est momentum variabilis fortuiti $\,X$ ordine $\,n$.

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

$m_{k}=\operatorname {E} (X^{k})={\frac {\operatorname {d} ^{k}}{\operatorname {d} t^{k}}}M_{X}(t){\biggr \vert }_{t=0}.$

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

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

Nexus interni

• Momentum (statistica)
• Fornix (statistica)