Module stats 

Symbolic probability and statistics: random variables, exact moments, probabilities, densities. Symbolic probability and statistics (SymPy’s stats): random variables with named distributions, exact moments, probabilities of events, densities and distribution functions — all as expressions — plus conditioning, transformations and mixtures of distributions.

A RandomVariable is a symbol together with a Distribution. The queries are exact where the distribution’s parameters are exact:

use symplex::prelude::*;
use symplex::stats::{Distribution, RandomVariable};

let ctx = Context::new();
let x = RandomVariable::new(&ctx, "X", Distribution::normal(ctx.int(0), ctx.int(1)));
assert_eq!(x.mean(), ctx.int(0));
assert_eq!(x.variance(), ctx.int(1));
// E[X² + 3X] for a standard normal.
assert_eq!(x.expectation(&(x.symbol().powi(2) + 3 * x.symbol())).simplify(), ctx.int(1));

let y = RandomVariable::new(&ctx, "Y", Distribution::binomial(ctx.int(5), ctx.rational(1, 3)));
assert_eq!(y.mean(), ctx.rational(5, 3));
assert_eq!(y.probability(&y.symbol().gt(&ctx.int(2)))?, ctx.rational(17, 81));
// Conditioning, and a transformed variable.
let half = x.given(&x.symbol().gt(&ctx.int(0)))?;          // X | X > 0
assert_eq!(half.mean().equals(&(ctx.int(2) / ctx.pi()).sqrt()), Some(true));
let w = x.transform("W", &(2 * x.symbol() + 1))?;          // 2X + 1 ~ Normal(1, 2)
assert_eq!(w.variance(), ctx.int(4));

Design

• A distribution is a Family: its support (Support, a finite union of intervals and points, continuous or on the integer lattice), its density (or probability mass function) as an expression in a free variable, and the closed forms it happens to have (moments, CDF, MGF, quantile, entropy), each optional. The generic machinery on Distribution answers every query from those: it clips an event’s region to the support and measures each piece through the CDF, or by exact integrate_definite / summation of the density, and it takes E[g(X)] through the raw moments when g is a polynomial. Implement Family to add a distribution of your own.
• Families compose: Truncated (conditioning), Affine (aX + b), Transformed (g(X) by the change-of-variables formula) and Mixture wrap other distributions and transport their closed forms exactly where the transport is exact.
• Parameters are expressions: rational parameters give exact rational answers, symbolic parameters give symbolic answers (E[X] = μ). Parameter validity (σ > 0, 0 ≤ p ≤ 1) is the caller’s promise for symbolic parameters and is checked for numeric ones by the try_ constructors.
• Nothing here is numerical by default: Distribution::sample is the only place a random number generator appears, and it is a plain deterministic-seed generator so tests are reproducible.

Distribution families: the structs re-exported below (Normal, Binomial, …); each has a Distribution::name(…) constructor and a try_name twin.

Modules

aggregation

Turning many raters’ labels into one answer: majority, plurality and weighted votes (exact), the Dawid–Skene EM model of rater confusion, Bradley–Terry strengths from pairwise comparisons, and the worker quality helpers that go with them (accuracy against gold labels, per-category precision / recall / F₁, confidence intervals for a proportion, gold-question screening).

agreement

Inter-rater agreement, exactly: percent agreement, Cohen’s κ (plain and weighted), Scott’s π, Fleiss’ κ, Krippendorff’s α, Gwet’s AC₁, the intraclass correlations of Shrout & Fleiss and Kendall’s W.

data

Descriptive statistics on observed data, exactly.

estimation

Parameter estimation from observed data: maximum likelihood, the method of moments, and Bayesian conjugate updating — every estimate handed back as a Distribution so it plugs into the rest of stats.

hypothesis

Hypothesis tests, effect sizes, multiple-comparison corrections, resampling and power / sample-size utilities.

information

Information theory on finite distributions, exactly: entropies, divergences and distances between probability vectors, and the information shared by the two margins of a joint table.

markov

Discrete-time Markov chains on a finite state space with an exact rational transition matrix (SymPy’s stats.DiscreteMarkovChain).

multivariate

The multivariate normal distribution with symbolic parameters, and exact multivariate descriptive statistics on data (covariance and correlation matrices, principal components).

order

Order statistics: the distribution of the k-th smallest of n independent draws from a distribution, as a Family of its own.

regression

Regression: exact ordinary and weighted least squares over ℚ, and logistic regression by Newton–Raphson in f64 (statsmodels’ OLS, WLS, Logit; numpy’s polyfit; scipy’s linregress).

reliability

Scale reliability, item analysis and further agreement / association statistics: Cronbach’s α and its relatives (standardized α, KR-20, split-half with the Spearman–Brown prophecy, α-if-deleted, Guttman’s λ₂), classical item analysis (difficulty, the D discrimination index, point-biserial and corrected item–total correlations), a confidence interval and a test for Cohen’s κ, κ_max, Cochran’s Q, the ordinal association measures (Goodman–Kruskal γ, Somers’ D, Stuart’s τ-c), contingency-table diagnostics (expected counts, χ² contributions, standardized and adjusted residuals) and inference on Pearson’s r (Fisher’s z, a confidence interval, the t-test, comparing two r’s).

sequential

Wald’s sequential probability ratio test (SPRT): decide between two simple hypotheses one observation at a time, stopping as soon as the evidence is strong enough — the tool for screening workers, raters or models on the fly rather than after a fixed batch.

survival

Time-to-event analysis with right censoring, exactly: how long until a respondent completes (or abandons) a task, how long a worker stays active, how long an item takes to reach agreement.

Structs

Affine

Y = aX + b for a ≠ 0 of known sign. Every closed form of X is transported exactly: E[Y] = aμ + b, Var[Y] = a²σ², raw moments by the binomial expansion, F_Y(y) = F_X((y−b)/a) (mirrored for a < 0), M_Y(t) = e^{bt} M_X(at), Q_Y(p) = aQ_X(p) + b (or aQ_X(1−p) + b), H[Y] = H[X] + ln|a| (continuous). SymPy: density(a*X + b).

Bernoulli

Bernoulli(p): P(X = 1) = p, P(X = 0) = 1 − p on {0, 1}.

Beta

Beta(α, β): density x^{α−1} (1−x)^{β−1} / B(α, β) on [0, 1].

Binomial

Binomial(n, p): P(X = k) = C(n, k) pᵏ (1−p)ⁿ⁻ᵏ on 0..=n.

Cauchy

Cauchy(x₀, γ): density 1 / (πγ (1 + ((x−x₀)/γ)²)) on ℝ. No moments of any order exist.

ChiSquared

ChiSquared(k) = Gamma(k/2, 2): density x^{k/2−1} e^{−x/2} / (2^{k/2} Γ(k/2)) on [0, ∞).

DiscreteUniform

DiscreteUniform(a, b): P(X = k) = 1/(b−a+1) on the integers a..=b. Die(s) is DiscreteUniform(1, s).

Distribution

A probability distribution: a shared handle to a Family with the generic exact machinery on top. Construct with the associated functions (Distribution::normal(…), binomial(…), …, each with a try_ twin validating numeric parameters), with the wrapper constructors (truncated, affine, transformed, mixture), or from your own family with from_family.

Exponential

Exponential(λ): density λ e^{−λx} on [0, ∞).

FDistribution

FDistribution(d₁, d₂): density √((d₁x)^{d₁} d₂^{d₂} / (d₁x + d₂)^{d₁+d₂}) / (x B(d₁/2, d₂/2)) on (0, ∞) (SymPy’s FDistribution(d1, d2)).

Finite

An explicit finite table P(X = vᵢ) = pᵢ (SymPy FiniteRV); the values need not be integers.

Gamma

Gamma(k, θ): density x^{k−1} e^{−x/θ} / (Γ(k) θᵏ) on [0, ∞) (shape–scale parametrisation, SymPy’s Gamma(k, theta)).

Geometric

Geometric(p): the number of trials up to and including the first success, P(X = k) = (1−p)^{k−1} p on 1..∞ (SymPy’s convention).

Hypergeometric

Hypergeometric(N, m, n): the number of successes in n draws without replacement from a population of N containing m successes, P(X = k) = C(m, k) C(N−m, n−k) / C(N, n) on max(0, n+m−N) ..= min(n, m).

Laplace

Laplace(μ, b): density e^{−|x−μ|/b} / (2b) on ℝ.

LogNormal

LogNormal(μ, σ): ln X ~ Normal(μ, σ); density e^{−(ln x − μ)²/(2σ²)} / (xσ√(2π)) on (0, ∞).

Logistic

Logistic(μ, s): density e^{−(x−μ)/s} / (s (1 + e^{−(x−μ)/s})²) on ℝ.

Mixture

A finite mixture Σ wᵢ Fᵢ of distributions of one kind. Every query is the weighted sum of the components’ answers, so nothing is lost when their supports overlap.

NegativeBinomial

NegativeBinomial(r, p): the number of failures before the r-th success, P(X = k) = C(k+r−1, k) pʳ (1−p)ᵏ on 0..∞ (SymPy’s convention).

Normal

Normal(μ, σ): density e^{−(x−μ)²/(2σ²)} / (σ√(2π)) on ℝ.

Pareto

Pareto(x_m, α): density α x_mᵅ / x^{α+1} on [x_m, ∞).

Poisson

Poisson(λ): P(X = k) = λᵏ e^{−λ} / k! on 0..∞.

RandomVariable

A random variable: a symbol with a distribution. Expressions in the symbol are random quantities; the queries below evaluate them exactly.

Rng

A deterministic pseudo-random generator (SplitMix64). Not cryptographic; seeded, reproducible, and good enough for Monte-Carlo sanity checks of exact results.

StudentT

StudentT(ν): density Γ((ν+1)/2) / (√(νπ) Γ(ν/2)) · (1 + x²/ν)^{−(ν+1)/2} on ℝ.

Support

A finite union of intervals and points on the line, with the Kind that says how a density over it is read.

Transformed

Y = g(X) for a continuous X and a g with explicit inverse branches: f_Y(y) = Σ_i f_X(h_i(y)) |h_i′(y)| (the change-of-variables formula), on the image of the support. Built by Distribution::transformed, which recognises a strictly monotone g on the support (one branch), and X², |X|, and even powers on a support symmetric enough for the two-branch formula. SymPy: density(g(X)).

Triangular

Triangular(a, b, c): density rising linearly from 0 at a to 2/(b−a) at the mode c and falling linearly to 0 at b, on [a, b] with a ≤ c ≤ b.

Truncated

The inner distribution conditioned on a region: density f(x)/P(R) on support ∩ R. Built by Distribution::truncated / RandomVariable::given. SymPy: given(X, cond).

Uniform

Uniform(a, b): density 1/(b − a) on [a, b].

Weibull

Weibull(λ, k): density (k/λ) (x/λ)^{k−1} e^{−(x/λ)ᵏ} on [0, ∞). SymPy’s Weibull(alpha, beta) has alpha = λ (scale) and beta = k (shape).

Enums

Kind

Whether a distribution puts its mass on intervals of ℝ (a density) or on isolated points (a probability mass function).

Piece

One connected part of a Support.

Traits

Family

A probability distribution family: its support and density, plus whatever closed forms it has (each defaults to “none”, which makes the generic machinery in Distribution integrate or sum instead).

Functions

conditional_expectation

E[g(X) | event] = E[g(X)·1_event] / P(event) for an event in the same variable: g · density integrated (summed) over the part of the support where the event holds, divided by its probability. SymPy: E(X, X > 0) (= √(2/π) for a standard normal), E(X**2, X > 0).

conditional_probability

P(event | given) = P(event ∧ given) / P(given) for two events in the same variable. SymPy: P(X > 1, X > 0) (= 2·P(X > 1) for a standard normal).

correlation

Pearson correlation Cov[g, h] / √(Var[g]·Var[h]). SymPy: correlation(X, 2*X + 1) (= 1).

covariance

Cov[g, h] = E[gh] − E[g]E[h] for expressions in independent variables. SymPy: covariance(X, 2*X) (= 2 for a standard normal X).

entropy

The (differential) entropy H[X] = −E[ln f(X)] in nats — for a discrete variable the Shannon entropy −Σ p(k) ln p(k). SymPy: entropy(X) (log(2)/2 + 1/2 + log(pi)/2 for a standard normal).

expectation

E[g(X₁, …, Xₖ)] for independent random variables. SymPy: E(X*Y), E(X + Y), E(expr) with several variables in expr.

probability

P(event) for an event in several independent variables. SymPy: P(And(X > 0, Y > 0)), P(X < Y).

same_family

The equality every family’s Family::eq_family delegates to: the other family is the same concrete type and compares equal.

sum_distribution

The distribution of A + B for independent a, b when it belongs to a closed family; None otherwise. The closure results implemented (parameters that must agree — p, λ — are compared structurally):

variance

Var[g] = E[g²] − E[g]² for an expression g in independent variables. SymPy: variance(X + Y), variance(2*X + 1).

Type Aliases

Sampler

A closure producing one f64 sample per call.