Reference Materials
Quick, citable reference sheets on probability distributions — history, definition, moments, parameter estimation, and simulation, all in one place per topic.
Continuous
26 distributionsNormal Distribution
X ~ N(μ, σ²)
The bell curve: history from de Moivre through Gauss and Laplace, the density and CDF, what location and scale mean geometrically, the full moment/entropy/hazard/quantile property list, maximum-likelihood estimation of the mean and variance, and how to simulate draws via the Box–Muller transform.
Exponential Distribution
X ~ Exp(λ)
The waiting-time distribution: history from Erlang's queueing theory and radioactive decay, the memoryless property, rate-vs-scale parameterization, the full property list (including its defining constant hazard function), maximum-likelihood estimation of the rate with its small-sample bias, and exact simulation via inverse-transform sampling.
Uniform Distribution
X ~ Uniform(a, b)
The flat-density distribution: history from Laplace's principle of indifference through the probability integral transform underlying every other simulation method in this series, order-statistic maximum-likelihood estimation (with no calculus involved), and why Fisher information doesn't apply in the usual sense when support depends on the parameters.
Gamma Distribution
X ~ Gamma(α, β)
The waiting-time-for-the-n-th-event distribution: history from Euler's Gamma function through Erlang's queueing generalization of the Exponential, shape-rate vs. shape-scale parameterization pitfalls, the full property list including the Fisher information matrix, maximum-likelihood estimation via a transcendental digamma equation, and simulation via the Erlang sum-of-Exponentials construction.
Beta Distribution
X ~ Beta(α, β)
The distribution for anything bounded to (0,1) -- a probability, proportion, or share: history from Euler's Beta function through Thomas Bayes's 1763 essay, the range of shapes (U-shaped, flat, skewed) the two parameters cover, exact ties to two already-published sheets (Beta(1,1) = Uniform, and the k-th order statistic of n iid Uniforms is exactly Beta(k, n-k+1)), its role as the Binomial's conjugate prior, closed-form method-of-moments plus numerical MLE, and simulation via a ratio of two independent Gammas.
Log-Normal Distribution
X ~ LogNormal(μ, σ²)
The distribution for skewed, always-positive data (sizes, incomes, concentrations): history from Galton and McAlister's 1879 multiplicative-effects argument, the density and CDF obtained directly from the already-published Normal sheet via X = e^Y where Y = ln X, the classic mu-and-sigma-squared-aren't-the-mean-and-variance pitfall, why the moment generating function provably does not exist despite every ordinary moment being finite, a fully closed-form MLE inherited unchanged from the Normal sheet, and simulation via exponentiating a Normal draw.
Weibull Distribution
X ~ Weibull(k, λ)
The reliability engineer's default lifetime model: history from Fréchet's extreme-value theory and Rosin-Rammler particle sizing through Waloddi Weibull's 1951 paper, a single shape parameter that spans decreasing, constant, and increasing hazard rates (the reliability 'bathtub curve'), a rare fully closed-form CDF and quantile function, a Fisher information matrix verified in closed form via the Euler-Mascheroni constant, and simulation via direct inverse-transform sampling.
Student's t-Distribution
X ~ t_ν
The small-sample fix for the Normal: history from William Sealy Gosset's 1908 Guinness brewery work under the pen name 'Student,' its exact Normal-over-Chi-Square construction, why heavier tails than the Normal account for the extra uncertainty of an estimated standard deviation, convergence to the standard Normal as degrees of freedom grow, why a moment generating function never exists despite finite low-order moments, a verified closed-form Fisher information via digamma and trigamma functions, and simulation via its own defining construction.
Chi-Square Distribution
X ~ χ²_k
The sum-of-squared-Normals distribution behind Pearson's 1900 goodness-of-fit test: history from Helmert's 1875-76 geodetic-error work, its exact identity as the already-published Gamma density with shape k/2 and rate 1/2, the full property list including a closed-form Fisher information via the trigamma function, a transcendental digamma maximum-likelihood equation verified against direct numerical optimization, and simulation via its own defining sum-of-squared-Normals construction.
F Distribution
X ~ F_(d1, d2)
The variance-ratio distribution behind every ANOVA and regression F-test: history from Fisher's 1924 variance-ratio work through George Snedecor's 1934 naming, its exact link to the already-published Beta distribution via a genuine change of variable, why no moment generating function exists, a verified closed-form 2x2 Fisher information matrix in the trigamma function, a two-dimensional transcendental maximum-likelihood system with no closed form, and simulation via its own defining ratio-of-Chi-Squares construction.
Laplace Distribution
X ~ Laplace(μ, b)
The 'two exponentials back to back' distribution: history from Pierre-Simon Laplace's 1774 work, a constant excess kurtosis of 3 regardless of scale (heavier-tailed than the Normal at every b), a median-based maximum-likelihood estimator that previews robust statistics, a direct tie to the Lasso's L1 penalty via its role as a Bayesian MAP prior, and simulation via the difference of two independent Exponentials.
Pareto Distribution
X ~ Pareto(x_m, α)
The 80/20-rule distribution: history from Vilfredo Pareto's 1896 land-ownership observation, the continuous counterpart to the already-published Zipf's law, a stark warning that the mean and variance can silently fail to exist depending on the tail index, a closed-form maximum-likelihood estimator for the shape parameter, and simulation via a log-transform of the Exponential.
Triangular Distribution
X ~ Triangular(a, c, b)
The three-number distribution behind PERT project scheduling and Monte Carlo risk analysis: a piecewise-linear density set directly by a minimum, mode, and maximum with no shape-fitting required, an exact sum-of-two-Uniforms construction that previews the Central Limit Theorem, and a mode with no closed-form maximum-likelihood estimator (verified via grid search) that nonetheless beats the closed-form method-of-moments estimator on efficiency.
Logistic Distribution
X ~ Logistic(μ, s)
The distribution behind logistic regression's sigmoid: history from Verhulst's 1830s-40s population-growth curve to its role as the latent-variable error distribution that makes logistic regression's P(Y=1|x)=sigmoid(xβ) formula exact (confirmed by simulation), the only continuous CDF on this site needing no special function, constant excess kurtosis of 6/5 echoing the already-published Laplace, and a transcendental maximum-likelihood fit mirroring the iterative solving logistic regression itself requires.
Half-Normal Distribution
X ~ HalfNormal(σ)
Defined directly as the absolute value of the already-published Normal: PDF/CDF verified against scipy.stats.halfnorm, the |Z| construction confirmed via 3-million-draw simulation, and a rare fully closed-form maximum-likelihood estimator (sigma-squared-hat equals the mean of the squared observations) verified against direct likelihood maximization -- a simplicity most sheets at this stage of the library don't have.
Folded Normal Distribution
X ~ FoldedNormal(μ, σ)
Generalizes the already-published Half-Normal by folding an off-center Normal(μ,σ²): PDF/CDF verified against scipy.stats.foldnorm, the μ=0 special case confirmed to collapse exactly onto the Half-Normal sheet's own mean and variance, and -- unlike its Half-Normal special case -- a maximum-likelihood fit with no closed form, verified via Nelder-Mead cross-checked against an independent 2D grid search.
Rayleigh Distribution
X ~ Rayleigh(σ)
The two-dimensional magnitude analogue of the already-published Half-Normal, and the standard model for received-signal amplitude in wireless propagation: PDF/CDF verified against scipy.stats.rayleigh, exact ties confirmed numerically to the already-published Weibull (k=2 special case) and Chi-Square/Exponential sheets, and -- like the Half-Normal -- a rare fully closed-form maximum-likelihood estimator.
Trapezoidal Distribution
X ~ Trapezoidal(a, b, c, d)
Sits directly between the already-published Uniform and Triangular distributions and contains both as exact limiting cases (confirmed numerically to 2.8e-17 agreement at the Triangular limit); mean and second-moment closed forms derived via symbolic integration; maximum likelihood of the two interior breakpoints has no closed form, verified via 2D grid search showing convergence to the true values as the sample size grows.
Arcsine Distribution
X ~ Arcsine(a, b)
One of Lévy's random-walk arcsine laws, and exactly the already-published Beta(1/2,1/2) on (0,1): PDF/CDF verified against scipy.stats.arcsine, the classic cos(Θ) circular-motion construction confirmed via 3-million-draw simulation. Has no formal MLE -- the likelihood is unbounded as the endpoints approach the sample min/max -- yet the sample min/max converge to the true endpoints at a verified O(1/n²) rate, a full order of n faster than the already-published Uniform's own endpoint estimator.
Skewed Normal Distribution
X ~ SkewNormal(ξ, ω, α)
Azzalini's one-parameter skew extension of the already-published Normal, exactly recovered at α=0: PDF/CDF verified against scipy.stats.skewnorm, and the delta-decomposition construction (which reuses the already-published Half-Normal via its |Z₀| term) confirmed via 3-million-draw simulation. Skewness is bounded, unlike the unbounded skewness of the already-published Weibull or Gamma. Maximum-likelihood fit has no closed form, verified via joint numerical optimization.
Generalized Normal Distribution
X ~ GenNormal(μ, α, β)
Also known as the Exponential Power distribution -- the cleanest demonstration on this site that three already-published distributions are one parameter apart: β=1 gives the already-published Laplace exactly, β=2 gives the already-published Normal exactly, and β→∞ converges toward the already-published Uniform (all three ties confirmed numerically). The excess-kurtosis formula evaluates to exactly 0 at β=2 and exactly 3 at β=1, matching those sheets' own reported values. Maximum likelihood of the shape parameter has no closed form.
Non-Central Chi-Square Distribution
X ~ χ²_k(λ)
The exact non-null distribution behind every chi-square test's power calculation: history from Fisher's 1920s treatment of the goodness-of-fit test's power, a Poisson-mixture density over the already-published Chi-Square's own densities with an exact λ=0 tie confirmed numerically, mean and variance formulas verified to ten decimal places, no closed-form maximum-likelihood estimator for either parameter, and simulation via the sum-of-squared-shifted-Normals construction that defines the distribution itself.
Non-Central F Distribution
X ~ F_(d1,d2)(λ)
The standard distribution behind ANOVA and regression F-test power calculations: a Poisson-mixture of rescaled already-published F densities built directly from the already-published non-central Chi-Square's own construction, with an exact λ=0 tie back to the already-published (central) F confirmed numerically, verified mean/variance formulas, no closed-form maximum-likelihood system, and simulation via a ratio of non-central to central Chi-Squares.
Non-Central t Distribution
X ~ t_ν(δ)
The distribution behind t-test power analysis and confidence intervals for Cohen's d: history from Jacob Cohen's power-analysis tradition, a shifted-Normal-over-Chi mixture integral verified against scipy.stats.nct to 3.7e-12, the first t-like sheet in this series where the classic symmetry deliberately breaks (nonzero skewness for δ≠0, with an exact δ=0 tie back to the already-published Student's t), no closed-form maximum-likelihood system, and simulation via its own defining construction.
Non-Central Beta Distribution
X ~ Beta(a, b, λ)
The exact distribution of sample R² under a nonzero population R², built the same way the already-published non-central F extends the already-published F: replacing the ratio's numerator Chi-Square with its non-central counterpart, giving a Poisson-mixture-of-Beta density with no scipy equivalent, verified against a truncated-mixture quadrature and a 3-million-draw simulation, no closed-form MLE for any of the three parameters, and simulation via the ratio-of-(noncentral, central)-Chi-Squares construction.
Kumaraswamy Distribution
X ~ Kumaraswamy(a, b)
The already-published Beta distribution's closed-form cousin: same two-shape-parameter flexibility on [0,1], but with an elementary CDF F(x)=1-(1-x^a)^b that inverts in closed form with no incomplete Beta function needed, making it the standard substitute for Beta in simulation studies and hydrological models, verified via exact agreement with numerical differentiation and a 5-million-draw simulation, no closed-form MLE despite the closed-form CDF, and simulation via a genuinely closed-form inverse-transform construction (no rejection sampling required, unlike Beta).
Discrete
18 distributionsBinomial Distribution
X ~ Binomial(n, p)
The count of successes in n independent trials: history from Bernoulli through de Moivre's normal approximation, the PMF and why discrete hazard/quantile functions are defined differently than in the continuous case, the full property list, maximum-likelihood estimation of p (exactly unbiased, exactly efficient), and simulation as a direct sum of Bernoulli draws.
Poisson Distribution
X ~ Poisson(λ)
The rare-event count distribution: history from Poisson's binomial limit through Bortkiewicz's horse-kick data, equidispersion (mean equals variance, and in fact every cumulant), the full property list, maximum-likelihood estimation of the rate, and exact simulation via Knuth's algorithm.
Geometric Distribution
X ~ Geometric(p)
The discrete analogue of the Exponential: history from the Pascal-Fermat correspondence through de Moivre, the only discrete distribution with the memoryless property, a constant discrete hazard function, maximum-likelihood estimation of the success probability (biased in finite samples), and inverse-transform simulation.
Negative Binomial Distribution
X ~ NegBinom(r, p)
The standard fix for overdispersed count data: history from Greenwood and Yule's 1920 accident-proneness study through the exact Gamma-Poisson mixture representation, why variance always exceeds the mean (and only approaches Poisson's equidispersion in a limit), maximum-likelihood estimation of the success probability, and simulation via a sum of Geometric draws.
Hypergeometric Distribution
X ~ Hypergeometric(N, K, n)
The distribution for sampling without replacement: history from early urn problems through Fisher's 1934 exact test for contingency tables, the exact convergence to the already-published Binomial as the population grows large relative to the sample, the finite-population-correction variance formula, the classic Lincoln-Petersen mark-recapture maximum-likelihood estimator (with a verified likelihood tie at its boundary case), and simulation via direct without-replacement draws.
Zipf Distribution
X ~ Zipf(N, s)
The rank-frequency power-law distribution behind Zipf's law: history from Pareto, Auerbach, and Estoup's independent early observations through George Zipf's 1949 'least effort' account, the infinite-support zeta-distribution limit, why the mode is always rank 1, the exponential-family structure that makes Fisher information exactly the variance of log-rank under the model itself, a maximum-likelihood estimator verified two independent ways, and simulation via inverse-transform sampling on the discrete CDF.
Discrete Uniform Distribution
X ~ DiscreteUniform(a, b)
The distribution of a fair die or any lottery drawing: history reaching back to Cardano and the Fermat-Pascal correspondence, the famous World War II 'German Tank Problem' application, why it is the maximum-entropy distribution over a fixed set of outcomes, why Fisher information doesn't apply for the same structural reason as the continuous Uniform, order-statistic maximum-likelihood estimation with no clean bias-correction formula, and direct simulation.
Bernoulli Distribution
X ~ Bernoulli(p)
The single yes/no trial underlying the entire success/failure family on this site: history from Jacob Bernoulli's posthumous 1713 Ars Conjectandi, its exact identity as the n=1 case of the already-published Binomial, why a fair coin (p=0.5) simultaneously maximizes variance and entropy, a fully closed-form maximum-likelihood estimator that exactly attains the Cramér-Rao bound, and direct simulation.
Beta-Binomial Distribution
X ~ BetaBinom(n, α, β)
The overdispersion fix for Binomial count data: history from Yule and Pearson's non-independent-trials work, the exact Beta-Binomial mixture representation that parallels the already-published Negative Binomial's Gamma-Poisson mixture, exact ties to two already-published sheets (α=β=1 recovers the Discrete Uniform; α,β→∞ recovers the ordinary Binomial), a Fisher information matrix expressed as a finite sum with no simpler closed form, a coupled two-parameter digamma maximum-likelihood system verified against direct optimization, and simulation via the Beta-mixing construction.
Skellam Distribution
K ~ Skellam(μ1, μ2)
The distribution of the difference between two independent Poisson counts: history from John Gordon Skellam's 1946 work, the first discrete sheet in this series with support over all integers rather than just the non-negative ones, a probability mass function built from the modified Bessel function, a closed-form method-of-moments estimator standing in for a maximum-likelihood equation with no closed form, and direct simulation as X1 - X2.
Logarithmic Distribution
K ~ Logarithmic(p)
Fisher's 1943 species-abundance model: history from Fisher, Corbet, and Williams's moth-trapping data at Rothamsted, the first discrete sheet in this series with support starting at 1 rather than 0, an exact compound-Poisson tie showing this is precisely the 'jump size' distribution hiding inside the already-published Negative Binomial, a maximum-likelihood equation with no closed form solved by one-dimensional root-finding, and direct simulation via NumPy's dedicated logseries generator.
Zero-Inflated Poisson Distribution
K ~ ZIP(π, λ)
Diane Lambert's 1992 fix for count data with more zeros than the Poisson predicts: history from manufacturing defect-count modeling, a two-component mixture of a 'structural' zero and an ordinary already-published Poisson process, a contrast with the already-published Negative Binomial's different (rate-heterogeneity) explanation for the same symptom of overdispersion, a maximum-likelihood system with no closed form verified against an independent statsmodels fit, and simulation via the mixture's own defining construction.
Negative Hypergeometric Distribution
X ~ NegHyper(N, K, r)
The missing corner of the 2×2 sampling-design table: draws without replacement until a fixed number of failures, filling in exactly where the already-published Hypergeometric (fixed sample size, without replacement) and Negative Binomial (fixed failures, with replacement) leave off, with a discrete-parameter maximum-likelihood estimator of the population's success count derived via the same ratio-of-likelihoods technique as the Hypergeometric sheet's own estimator, and simulation via the distribution's own without-replacement draw.
Zero-Inflated Negative Binomial Distribution
X ~ ZINB(π, r, p)
Two overdispersion mechanisms stacked in one model: the already-published Zero-Inflated Poisson's structural-zero mixture applied on top of the already-published Negative Binomial instead of the Poisson, reached for when a fitted Zero-Inflated Poisson still under-predicts the tail, with a variance formula derived via the law of total variance on a Bernoulli-times-NegBinom product, a no-closed-form maximum-likelihood system verified against an independent method-of-moments fit, and simulation via the mixture's own defining construction.
Yule-Simon Distribution
X ~ YuleSimon(ρ)
The rich-get-richer distribution: history from Udny Yule's 1925 model of species-per-genus growth and Herbert Simon's 1955 rediscovery via word-frequency preferential attachment, a genuinely infinite-support power-law tail whose exponent is derived from the generative process itself (unlike the already-published Zipf sheet's assumed finite-support power law), moments that only exist up to a point set by ρ, a digamma-based maximum-likelihood equation with no closed form, and simulation via an exact Beta-Geometric mixture representation.
Conway-Maxwell-Poisson Distribution
X ~ CMP(λ, ν)
The one distribution in this library that moves in either direction from the Poisson's Var = E baseline through a single dispersion parameter: history from Conway and Maxwell's 1962 state-dependent queueing model to its 2005 popularization for count regression, an intractable normalizing constant with no closed form except at the Poisson-recovering ν=1 case, large-λ asymptotic mean/variance approximations checked against direct summation, and simulation via truncated inverse-CDF sampling.
Poisson-Binomial Distribution
X ~ PoiBin(p_1, ..., p_n)
The already-published Binomial distribution's heterogeneous generalization: a sum of independent Bernoulli trials that no longer share one probability, the exact distribution behind vote-count forecasting, portfolio credit risk, and reliability engineering, computed via an O(n²) dynamic-programming convolution verified against brute-force enumeration to machine precision, a proof that heterogeneous p_i always concentrates the sum more than a same-mean Binomial (Jensen's inequality on p(1-p)), a worked explanation of why individual p_i are not identifiable from repeated sums even though the aggregate mean and variance are, and simulation as a direct sum of independent Bernoulli draws.
Zipf-Mandelbrot Distribution
X ~ ZM(N, q, s)
Mandelbrot's plateau-correcting generalization of the already-published Zipf distribution: a single shift parameter q flattens the top-rank probabilities to fix Zipf's well-documented underfit of real word-frequency data, while leaving the power-law tail slope unchanged, verified via exact q=0 collapse to Zipf and a 2-million-draw simulation, a demonstrated weak joint identifiability between q and s that makes maximum-likelihood recovery visibly less tight than Zipf's own single-parameter fit, and simulation via the same inverse-transform construction used on the Zipf sheet.
Multivariate
7 distributionsMultinomial Distribution
X ~ Multinomial(n, p)
The first multivariate sheet in this series -- the direct multi-category generalization of the already-published Binomial behind Pearson's chi-square test: an exact construction as independent Poisson counts conditioned on a fixed sum, the full covariance structure showing every pair of categories is negatively correlated, a verified (k-1)x(k-1) Fisher information matrix, a rare fully closed-form maximum-likelihood estimator (the sample proportion in each category), and simulation via the Poisson-conditioned-on-sum construction.
Multivariate Normal Distribution
X ~ N_p(μ, Σ)
The joint generalization of the Normal distribution underlying ordinary linear regression and the multivariate trinity of tests: history from Galton and Pearson's correlation theory, an exact Schur-complement formula for conditional distributions verified via Bayes' rule to six decimal places, the Mahalanobis distance's exact tie to the already-published Chi-Square sheet, a rare case where zero covariance genuinely implies independence, and simulation via the Cholesky construction.
Dirichlet Distribution
X ~ Dirichlet(α)
The conjugate prior for a probability vector, and the direct multivariate generalization of the already-published Beta distribution: every marginal and every aggregated grouping of categories is itself Beta or Dirichlet, an exact posterior-update tie to the already-published Multinomial sheet, a maximum-likelihood system with no closed form requiring digamma-based fixed-point iteration, and simulation via a ratio of independent Gamma draws.
Wishart Distribution
W ~ Wishart_p(n, Σ)
The first random-matrix sheet on the site: John Wishart's 1928 sampling distribution for an estimated covariance matrix, built directly as a sum of outer products of already-published Multivariate Normal draws, with an exact p=1 collapse onto the already-published Chi-Square confirmed numerically. A rare closed-form maximum-likelihood estimator for the scale matrix, and simulation via both the defining construction and the more efficient Bartlett decomposition.
Inverse Wishart Distribution
X ~ InvWishart_p(ν, Ψ)
The standard conjugate prior for a Multivariate Normal's covariance matrix, defined directly as the matrix inverse of the already-published Wishart (exact p=1 collapse onto the already-published Inverse Gamma, confirmed numerically). Every diagonal entry is exactly Inverse-Gamma distributed regardless of the rest of the scale matrix's structure (confirmed via Kolmogorov-Smirnov test), and the posterior-conjugacy update was verified exactly to machine precision rather than only by simulation.
Multivariate t Distribution
X ~ t_p(ν, μ, Σ)
The direct joint generalization of the already-published Student's t, built by dividing an already-published Multivariate Normal by a single shared Chi-Square scale (confirmed numerically to match scipy.stats.multivariate_t exactly, and to converge to the Multivariate Normal as ν grows). Every marginal is a univariate t sharing the same degrees of freedom -- verified via a Kolmogorov-Smirnov test -- and the covariance is always larger than the scale matrix Σ itself, exactly as in the univariate case.
Multivariate Hypergeometric Distribution
X ~ MVHypergeom(N, K, n)
The missing without-replacement counterpart to the already-published Multinomial, and the direct multi-category generalization of the already-published Hypergeometric: draws without replacement from a population split into k labeled categories. Confirmed numerically to converge to the Multinomial as the population grows large relative to the sample, with a closed-form method-of-moments estimator for the category sizes standing in for a discrete, no-closed-form maximum-likelihood search.
Lifetime / Survival
4 distributionsLog-Logistic Distribution
X ~ LogLogistic(α, β)
The first sheet to open the site's Lifetime/Survival category on its own terms: defined by exponentiating the already-published Logistic distribution (exact tie confirmed via Kolmogorov-Smirnov test), with a signature 'proportional odds' survival property and a hazard that can rise and then fall -- a shape neither the already-published Weibull nor Exponential can produce. Verified against scipy.stats.fisk to machine precision, with no closed-form maximum-likelihood estimator.
Gompertz Distribution
X ~ Gompertz(b, η)
Benjamin Gompertz's 1825 'law of human mortality': an exponentially increasing hazard function that remains the actuarial industry's standard adult-mortality model two centuries later. Defined directly by its hazard rather than its density, verified against scipy.stats.gompertz to within 7e-17, with no closed-form maximum-likelihood estimator and simulation via a closed-form inverse-CDF transform.
Generalized Gamma Distribution
X ~ GenGamma(a, c, β)
E.W. Stacy's 1962 unifying capstone for this Lifetime/Survival batch, playing the same role the already-published Generalized Normal played for the Laplace/Normal/Uniform: a=1 gives the already-published Weibull exactly, c=1 gives the already-published Gamma exactly, and a=c=1 gives the already-published Exponential exactly (all three ties confirmed to 2.8e-17). No closed-form maximum-likelihood estimator for any of the three parameters.
Inverse Gaussian (Wald) Distribution
X ~ InvGaussian(μ, λ)
Defined by its probabilistic origin as the first-passage time of a drifting Brownian motion to a fixed level, and named for a cumulant-generating-function relationship to the already-published Normal rather than a literal reciprocal. A rare fully closed-form maximum-likelihood estimator, a confirmed Central-Limit-like convergence to the Normal as λ grows relative to μ, and simulation via the efficient Michael-Schucany-Haas algorithm verified against scipy.stats.invgauss to within 5.6e-17.
Heavy-Tailed
3 distributionsCauchy Distribution
X ~ Cauchy(x₀, γ)
The standard counterexample to averaging: history from Poisson's 1824 encounter with its Law-of-Large-Numbers-defying tails through Cauchy's use of it to popularize the pathology, exact ties to the already-published Student's t at one degree of freedom and to the ratio of two independent standard Normals, a mean and variance that provably do not exist (confirmed by a sample mean that never converges no matter the sample size), and a maximum-likelihood fit that beats the sample median with the textbook 8/π² asymptotic relative efficiency.
Lévy Distribution
X ~ Levy(μ, c)
The most extreme of the three stable distributions with elementary densities (alongside the already-published Normal and Cauchy): a full step beyond the Cauchy's undefined mean, since neither the mean nor the variance exists for any parameter value, verified via the exact reciprocal-of-squared-Normal construction (X = μ + c/Z²) confirmed by a Kolmogorov–Smirnov test against direct sampling, a closed-form median and characteristic function, a closed-form maximum-likelihood estimator for the scale given a known location, and a running-sample-mean demonstration of the divergence.
Generalized Pareto Distribution
X ~ GPD(μ, σ, ξ)
The distribution behind peaks-over-threshold extreme value theory: history from the Pickands–Balkema–de Haan theorem, a shape parameter ξ that continuously interpolates between the already-published Pareto's heavy unbounded tail (ξ>0), the already-published Exponential's light unbounded tail (ξ=0), and a hard finite upper bound (ξ<0), confirmed end-to-end by simulating exceedances of the already-published Pareto over a high threshold and fitting a GPD to them, no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
Bayesian / Prior
3 distributionsInverse Gamma Distribution
X ~ InvGamma(α, β)
The standard conjugate prior for a Normal distribution's variance: defined directly as the reciprocal of the already-published Gamma (confirmed via 3-million-draw simulation), moments that exist only up to order α (confirmed by watching higher moments numerically diverge right at the cutoff), no moment generating function due to a polynomial right tail, a maximum-likelihood fit derived by reusing the Gamma sheet's own digamma equation on reciprocal data, and a Normal-variance posterior conjugacy confirmed to 13 decimal places against a brute-force grid computation.
Normal-Inverse-Gamma Distribution
(μ, σ²) ~ NIG(m, λ, a, b)
The first fully joint conjugate prior on this site: the complete Bayesian prior for a Normal's mean and variance together (extending the already-published Inverse Gamma's known-mean-only case), built hierarchically from the already-published Inverse Gamma and Normal sheets, with closed-form InvGamma and non-standardized Student's-t marginals verified against scipy, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own two-stage hierarchical construction.
Normal-Inverse-Wishart Distribution
(μ, Σ) ~ NIW(m, λ, ν, Ψ)
The multivariate capstone tying together four already-published sheets at once: the complete joint conjugate prior for a Normal's mean vector and covariance matrix together, generalizing the already-published Normal-Inverse-Gamma one dimension higher via the already-published Wishart and Inverse Wishart, with closed-form InvWishart and multivariate-t marginals, exact machine-precision confirmation of the closed-form posterior-update conjugacy, and simulation via its own hierarchical construction -- the standard prior underneath Bayesian multivariate regression and Gaussian mixture models.
Specialized
4 distributionsVon Mises Distribution
Θ ~ VM(μ, κ)
The site's first circular/directional distribution -- the natural analogue of the already-published Normal once a straight line is replaced by a circle: an exponential-family density exp(κcos(θ-μ)) normalized by a modified Bessel function, exact uniform-circle and large-κ Normal-approximation limits confirmed numerically, closed-form MLE for the mean direction (the sample circular mean) with a root-finding MLE for concentration, and simulation via the Best-Fisher rejection algorithm verified by a Kolmogorov-Smirnov test.
Gumbel Distribution
X ~ Gumbel(μ, β)
The light-tailed member of the Fisher-Tippett extreme-value family and the classic model for the maximum of many independent light-tailed observations: history from Gumbel's own flood/drought work, a max-of-Exponentials convergence result confirmed by simulation, a rare fully closed-form CDF, a clean exact tie to the already-published Exponential (e^-X for X~Gumbel(0,1)), no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
Generalized Extreme Value Distribution
X ~ GEV(μ, σ, ξ)
The block-maxima unifying capstone of extreme value theory, twin to the already-published Generalized Pareto's threshold-exceedance theory: one shape parameter ξ spans the already-published Gumbel (ξ=0), the heavy-tailed Fréchet (ξ>0), and a bounded Weibull-type shape (ξ<0), confirmed end-to-end via the Fisher-Tippett-Gnedenko convergence of normalized block maxima of the already-published Pareto to the theoretical Fréchet-type GEV, no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
Tweedie Distribution
Y ~ Tw_p(μ, φ)
The exponential dispersion family unifying five already-published shapes as exact special cases of a single mean-variance power law Var(Y)=φμ^p: the already-published Normal (p=0), Poisson (p=1), the already-published Gamma (p=2), and the already-published Inverse Gaussian (p=3), with the actuarially important 1<p<2 range giving a genuinely new compound-Poisson-Gamma shape with a point mass at zero plus a continuous density, verified via an infinite-series density formula matched exactly against a reference implementation, profile-likelihood parameter estimation, and simulation via the compound Poisson-Gamma construction.