Functio momenta generans
Functio momenta generans M X : R → R + ,   t ↦ M X ( t ) , {\displaystyle M_{X}\colon \mathbb {R} \to \mathbb {R} ^{+},\ t\mapsto M_{X}(t),} variabilis fortuiti X {\displaystyle \,X} sive distributionis eius exsistit, si pro omnibus n ∈ N {\displaystyle n\in \mathbb {N} } valores medii exspectati E ⁡ ( X n ) {\displaystyle \operatorname {E} (X^{n})} exsistunt. Tum M X {\displaystyle \,M_{X}} sic definitur: M X ( t ) = E ⁡ ( e t X ) . {\displaystyle M_{X}(t)=\operatorname {E} (\operatorname {e} ^{tX}).} M X {\displaystyle \,M_{X}} ad momenta calculanda adhibetur, quod expansione functionis exponentialis in seriem demonstratur: M X ( t ) = E ⁡ ( e t X ) = E ( ∑ n = 0 ∞ ( t X ) n n ! ) = ∑ n = 0 ∞ t n n ! E ⁡ ( X n ) = ∑ n = 0 ∞ t n n ! m n , {\displaystyle \,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 = E ⁡ ( X n ) {\displaystyle m_{n}=\operatorname {E} (X^{n})} est momentum variabilis fortuiti X {\displaystyle \,X} ordine n {\displaystyle \,n} . Haec expansio dicit quomodo momentum m k {\displaystyle \,m_{k}} omnis ordinis k ∈ N {\displaystyle k\in \mathbb {N} } calculari potest: m k = E ⁡ ( X k ) = d k d ⁡ t k M X ( t ) | t = 0 . {\displaystyle m_{k}=\operatorname {E} (X^{k})={\frac {\operatorname {d} ^{k}}{\operatorname {d} t^{k}}}M_{X}(t){\biggr \vert }_{t=0}.} Si E ⁡ ( X ) {\displaystyle \operatorname {E} (X)} parametro μ {\displaystyle \,\mu } familiae distributionum correspondet, momenta centralia eis momentis non-centralibus correspondent, quae ad parametrum μ = 0 {\displaystyle \,\mu =0} pertinent. Semper momenta centralia, μ k = E ⁡ ( [ X − E ⁡ ( X ) ] k ) {\displaystyle \mu _{k}=\operatorname {E} ([X-\operatorname {E} (X)]^{k})} , etiam ex relationibus ad momenta non-centralia, m k = E ⁡ ( X k ) {\displaystyle m_{k}=\operatorname {E} (X^{k})} , calculari possunt.