Conditioned

$a$ conditioned on $b$ — meaningful only as Probability's or Expectation's first argument.

Probability(Conditioned(pred, cond), Distributed(x, dist))P(pred | cond) = P(pred, cond) / P(cond), for the discrete and closed-form-CDF cases Probability itself handles; otherwise stays unevaluated.

Expectation(Conditioned(f, cond), Distributed(x, dist))E[f | cond]: exact for cond pinning x to a point (any distribution), or cond bounding x to a finite interval on a discrete distribution; otherwise stays unevaluated.

Typed signatures

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

Domain: Statistics

Details

• Conditioned is a bare inert constructor (like Distributed) — it does nothing on its own; both evaluations above happen inside Probability/Expectation, triggered by seeing Conditioned as their first argument.
• Probability's case is built entirely by calling back into Probability itself: the numerator is Probability(And(pred, cond), binding), the denominator Probability(cond, binding) — so it inherits every discrete-PDF and closed-form-CDF case any wave of this package has added, with no new distribution-shape knowledge needed here. One extra case past the plain And machinery: pred = Equal(x, k) is evaluated directly as PDF(k)/P(cond) (or 0, if cond fails at k), since Probability's own And case only recognizes a conjunction of two Less/LessEqual relations, not Equal combined with anything.
• Expectation's case: cond = Equal(x, k) collapses f to its value at k, for any distribution. A finite interval condition on a DISCRETE distribution sums exactly over the (necessarily finite) support the interval implies. A continuous distribution with an interval condition — which the exact machinery for a truncated continuous mean already exists for, in TruncatedDistribution — is NOT composed through here; out of scope for this batch, so it stays unevaluated.
• Neither case falls back to numeric approximation — see NProbability/NExpectation for that.

Examples

Basic #

$P(X{=}2 \mid X \le 5) = PDF(2)/CDF(5) = 45/184 \approx 0.2445652174$ for $X \sim \mathrm{Poisson}(3)$ — exact, the $e^{-3}$ factor cancels. Cross-checked against wolframscript's N[Probability[x == 2 \[Conditioned] x <= 5, x \[Distributed] PoissonDistribution[3]], 10].

live ↗

In$\mathrm{Probability}(\mathrm{Conditioned}(x=2, x≤5), \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$\frac{45}{184}$

$E[X \mid X \le 3] = 51/26 \approx 1.961538462$ for $X \sim \mathrm{Poisson}(3)$ — an exact finite sum over $\{0,1,2,3\}$. Cross-checked against wolframscript's N[Expectation[x \[Conditioned] x <= 3, x \[Distributed] PoissonDistribution[3]], 10].

live ↗

In$\mathrm{Expectation}(\mathrm{Conditioned}(x, x≤3), \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$1.961\,538\,461\,538\,462$

Conditioning on an exact point collapses $f$ to its value there, for any distribution.

live ↗

In$\mathrm{Expectation}(\mathrm{Conditioned}(x^2, x=3), \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$9$

Scope #

$P(X \le 3 \mid X \ge 1) \approx 0.6287483579$ for $X \sim \mathrm{Poisson}(3)$ — both pred and cond are single LessEqual relations, the shape Probability's own And case composes. Cross-checked against wolframscript's N[Probability[x <= 3 \[Conditioned] x >= 1, x \[Distributed] PoissonDistribution[3]], 10].

live ↗

In$\mathrm{Probability}(\mathrm{Conditioned}(x≤3, 1≤x), \mathrm{Distributed}(x, \operatorname{PoissonDistribution}(3)))$Out$\frac{12}{(1-1/e^3)e^3}$

$P(X \le 0 \mid X \ge -1) = (\Phi(0)-\Phi(-1))/(1-\Phi(-1)) \approx 0.4057132913$ for standard normal $X$ — the closed-form-CDF case, not just discrete PDFs. Cross-checked against wolframscript's N[Probability[x <= 0 \[Conditioned] x >= -1, x \[Distributed] NormalDistribution[0,1]], 10].

live ↗

In$\mathrm{Probability}(\mathrm{Conditioned}(x≤0, -1≤x), \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1)))$Out$-\frac{\operatorname{erf}(-\frac{2^{1/2}}{2})}{1-\operatorname{erf}(-2^{1/2}/2)}$

Possible issues #

A continuous distribution with an interval condition is out of scope here (see the truncated-mean note above) and stays unevaluated, even though Probability's analogous case (previous example) works.

live ↗

In$\mathrm{Expectation}(\mathrm{Conditioned}(x, x≤3), \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1)))$Out$\mathrm{Expectation}(\mathrm{Conditioned}(x, x≤3), \mathrm{Distributed}(x, \operatorname{NormalDistribution}(0, 1)))$

About

Examples	6

See also: Probability, Expectation, NProbability, NExpectation, TruncatedDistribution