SliceDistribution

The ordinary distribution answering a random process's value at a fixed time $t$.

SliceDistribution(proc, t)evaluates directly to the distribution WienerProcess or PoissonProcess answers at time $t$ — PDF/CDF/Mean/Variance need no changes of their own to reach it, since the slice IS already a plain NormalDistribution or PoissonDistribution.

Typed signatures

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

Domain: Statistics

Details

• proc(t), Wolfram's proc[t], is SliceDistribution(proc, t): the call reads as the slice.
• $SliceDistribution(WienerProcess(\mu,\sigma), t) = NormalDistribution(\mu t, \sigma\sqrt{t})$; $SliceDistribution(PoissonProcess(\lambda), t) = PoissonDistribution(\lambda t)$.
• Unevaluated for any process kind other than WienerProcess/PoissonProcess — no other process is declared yet.

Examples

Basic #

live ↗

In$\mathrm{SliceDistribution}(\mathrm{WienerProcess}(1, 2), 3)$Out$\operatorname{NormalDistribution}(3, 2\sqrt{3})$

live ↗

In$\mathrm{SliceDistribution}(\mathrm{PoissonProcess}(5), 2)$Out$\operatorname{PoissonDistribution}(10)$

Scope #

PDF/CDF/Mean/Variance need no changes of their own — the slice is already a plain PoissonDistribution.

live ↗

In$\mathrm{mean}(\mathrm{SliceDistribution}(\mathrm{PoissonProcess}(5), 2))$Out$10$

About

Examples	3

See also: WienerProcess, PoissonProcess, RandomFunction, PDF, Mean, Variance