Shape • Parameters • Moments
Explore Probability
Distributions
Learn the shape, parameters, moments, and real-world uses of common and advanced probability distributions through clear, visual explanations.
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Featured distributions
Compare foundational models for continuous and discrete random variables.
Showing 1–6 of 65 distributions
Normal Distribution
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
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.
Binomial Distribution
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
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.
Uniform Distribution
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
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.
Map of Probability Distributions
See how these distributions relate to each other — special cases, limiting cases, transformations, mixtures, and conjugate priors — across 8 focused panels.
Compare distributions side by side
Pick 2-4 distributions, adjust their parameters, and see how their shapes and moments compare in real time.