NExpectation

A numeric-only $E[f(x)]$ — falls back to summation/quadrature where Expectation stays symbolic.

NExpectation(f, Distributed(x, dist))tries the exact Expectation first; otherwise sums (discrete) or integrates (continuous) numerically. Always returns a machine float.

Typed signatures

Type	Package	Applies
(any, expression<Distributed>) -> number	statistics	always

Domain: Statistics

Details

• Tries Expectation(f, Distributed(x, dist)) first — a real (exact) answer whenever it resolves, just converted to a float. Only falls back to a numeric approximation past Expectation's own reach: f past a linear/quadratic polynomial in x, or a distribution Expectation has no closed form for at all.
• Discrete, finite support (e.g. BinomialDistribution): an exact finite sum, converted to a float.
• Discrete, infinite support (e.g. PoissonDistribution): summed with an infinite-tail cutoff — stops once the cumulative PDF mass is within $10^{-12}$ of $1$ and the current term is below $10^{-12}$.
• Continuous: double-exponential (tanh-sinh) quadrature, generalized to a finite interval, a semi-infinite one, or the whole real line depending on the distribution's domain, converging to the same $10^{-12}$ relative tolerance.
• Distributions this package has no tabulated support/domain for (including every wave-4 compound distribution — TruncatedDistribution, MixtureDistribution, etc.) stay unevaluated — a documented gap, not a bug.

Examples

Basic #

$E[\sin X] = \sin(\mu)e^{-\sigma^2/2}$ for $X \sim N(2,3)$; $\approx 0.01010138197$ — cross-checked against wolframscript's N[Expectation[Sin[x], x \[Distributed] NormalDistribution[2,3]], 10]. Expectation itself stays unevaluated here (Sin is past its linear/quadratic reach).

live ↗

In$\mathrm{NExpectation}(\sin(x), \mathrm{Distributed}(x, \operatorname{NormalDistribution}(2, 3)))$Out$0.010\,101\,381\,966\,839\,188$

$E[X^3] = 6/\lambda^3 = 6/8 = 0.75$ exactly, up to quadrature round-off; cross-checked against wolframscript's N[Expectation[x^3, x \[Distributed] ExponentialDistribution[2]], 10].

live ↗

In$\mathrm{NExpectation}(x^3, \mathrm{Distributed}(x, \operatorname{ExponentialDistribution}(2)))$Out$0.750\,000\,000\,000\,000\,1$

Scope #

$E[1/(X{+}1)] = (1-e^{-\mu})/\mu \approx 0.3167376439$ for $X \sim \mathrm{Poisson}(3)$ — an infinite discrete sum with a tail cutoff, cross-checked against wolframscript's N[NExpectation[1/(x+1), x \[Distributed] PoissonDistribution[3]], 10].

live ↗

In$\mathrm{NExpectation}(\frac{1}{x+1}, \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$0.316\,737\,643\,877\,377\,7$

A finite discrete sum over $\{0,\ldots,5\}$; cross-checked against wolframscript's N[NExpectation[x^3, x \[Distributed] BinomialDistribution[5, 2/5]], 10].

live ↗

In$\mathrm{NExpectation}(x^3, \mathrm{Distributed}(x, \operatorname{BinomialDistribution}(5, \frac{2}{5})))$Out$15.44$

About

Examples	4

See also: Expectation, NProbability, Distributed