CharacteristicFunction

The characteristic function $\varphi(t) = E[e^{itX}]$ of $dist$.

CharacteristicFunction(dist, t)a per-distribution closed form for a scoped subset of the distributions this package declares — see details. Every other distribution stays unevaluated.

Typed signatures

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

Domain: Statistics

Details

• Declared for: NormalDistribution, UniformDistribution, DiscreteUniformDistribution, BernoulliDistribution, BinomialDistribution, GeometricDistribution, NegativeBinomialDistribution, PoissonDistribution, GammaDistribution, ChiSquareDistribution, ErlangDistribution, LaplaceDistribution, CauchyDistribution (CF only — its MomentGeneratingFunction does not exist), and LogisticDistribution.
• Every distribution NOT in that list (Beta, StudentT, Weibull, LogNormal, Rayleigh, Pareto, Chi, HalfNormal, Maxwell, Triangular, Hypergeometric, Binormal, Empirical) is left undeclared here rather than faked — most need a special function (confluent hypergeometric, Bessel K) this package doesn't carry.
• $\varphi(0) = 1$ always; the uniform/discrete-uniform/logistic formulas special-case $t=0$ to avoid a removable $0/0$.

Plot

argument dist of type distribution has no control

Examples

Basic #

live ↗

In$\mathrm{CharacteristicFunction}(\operatorname{NormalDistribution}(2, 3), t)$Out$e^{2i t-\frac{9t^2}{2}}$

live ↗

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

live ↗

In$\mathrm{CharacteristicFunction}(\mathrm{CauchyDistribution}(0, 1), t)$Out$e^{-\vert t\vert}$

About

Examples	3

See also: MomentGeneratingFunction, PDF, Moment, Cumulant