MomentGeneratingFunction

The moment generating function $M(t) = E[e^{tX}]$ of $dist$.

MomentGeneratingFunction(dist, t)a per-distribution closed form, the same scoped subset CharacteristicFunction covers, minus CauchyDistribution (its MGF genuinely does not exist — $E[e^{tX}]$ diverges for any $t\neq0$) and plus LogisticDistribution via the Beta function.

Typed signatures

Type	Package	Applies
(distribution, complex) -> complex	statistics	always

Domain: Statistics

Details

• $M(t) = \varphi(t/i)$ in principle; this file writes each distribution's formula directly (the naive i -> 1 substitution of CharacteristicFunction's formula) rather than deriving one from the other, EXCEPT LogisticDistribution, whose MGF is $e^{\mu t}B(1-\beta t, 1+\beta t)$ — a different (still elementary) closed form from its CF.
• $M(0) = 1$ always; a numeric derivative of $M$ at $t=0$ reproduces Mean (checked in tests/distributions-3.test.ts, not repeated here).

Examples

Basic #

live ↗

In$\mathrm{MomentGeneratingFunction}(\mathrm{GammaDistribution}(2, 1), t)$Out$\frac{1}{(1-t)^2}$

live ↗

In$\mathrm{MomentGeneratingFunction}(\operatorname{PoissonDistribution}(2), 0)$Out$1$

Possible issues #

live ↗

In$\mathrm{MomentGeneratingFunction}(\mathrm{CauchyDistribution}(0, 1), 1)$Out$\mathrm{MomentGeneratingFunction}(\mathrm{CauchyDistribution}(0, 1), 1)$

About

Examples	3

See also: CharacteristicFunction, PDF, Moment, Cumulant