Distribution Lab shows probability distributions interactively — as preparation for GLMs and Bayesian models in brms. Which distribution fits which data? For each distribution: description, use cases, parameter explanation, brms usage and technical details. Category tabs (top) select the distribution group.
Distribution buttons below select the specific distribution.
Sliders on the left adjust parameters in real time. Mean (blue), Median (green), Mode (orange) as vertical dashed lines. Each moment independently toggleable. This is the most common stumbling block — and it has a good reason.

For one and the same distribution, statistics has several equivalent ways of writing it down. The Gamma distribution, for instance, can be described via (shape, rate), via (shape, scale), or via (mean, shape) — the same distribution, just different "dials". Which one you pick is pure convention.

Paul Bürkner, the author of brms, deliberately took a different route than most: he asked what users actually want to predict in a regression model — and the answer is almost always the mean. So he parametrised every distribution that can serve as a likelihood so that the regression's target is always μ. Whether Gamma, Beta, Weibull or Negative Binomial: y ~ x always predicts the mean. That's a sensible, user-friendly unification — you don't have to relearn what the first parameter means for every new distribution family.

Because this μ-parametrisation is central to working with brms, it's the default throughout this tool. For comparison, the toggle also lets you show the classical textbook notation — the one you'll meet in textbooks, in R's base functions (dgamma, dnbinom), and in publications.

So where does brms use the classical notation? Precisely whenever the distribution is not a likelihood but plays a different role — in particular as a prior. In prior(gamma(2, 0.1), ...) the two numbers are shape and rate, not μ and φ. That's internally consistent (a prior doesn't predict a mean — it directly describes the distribution of a parameter), but it can be confusing — especially for distributions like Skew-Normal, where both notations appear under the same function name.

The colour coding helps keep them apart: in the "What do the parameters mean?" section, the teal box 1 always stands for the brms/likelihood parametrisation, the wine-red box 2 for the classical notation or prior usage. The sidebar toggle uses the same colours.

The "Usage in brms" section then shows both roles concretely for each distribution: as likelihood (family syntax, link) and as prior (prior() syntax) — including typical parametrisations. ZI-Beta and ZOIB: point masses at 0/1 as red bars, Beta component as curve. Hurdle-Gamma and Hurdle-Lognormal: point mass at 0 as a red bar, positive component as a curve on (0, ∞) instead of [0,1]. Bar chart of the PMF. Mode = tallest bar. X locked / X free button (above the chart): Freezes the x-axis so that when changing parameters (e.g. μ or σ) the shape change is directly visible — instead of the axis rescaling. Active by default, and the on/off setting is global across all distributions. The frozen axis range itself, however, is remembered per distribution rather than shared — so continuous-unbounded and continuous-bounded distributions still each get their own sensible axis. The tool likewise remembers each distribution's last-set parameter values, so you can jump back and forth between distributions without losing your settings — until the page is reloaded, after which everything reverts to defaults. Disabled for discrete, mixed, and ordinal distributions (not meaningful there). ⊞ Relations button opens a large overview map with all distributions of the Distribution Lab and their connections: special cases, limits, constructions. Orientation in the distribution zoo. Inside it, 📖 Symbols expands a quick-reference for the Greek letters used. 📋 Cheat Sheet button opens a full reference: every symbol (name, pronunciation, note), the parameter meaning of every distribution grouped by category, and the arrow/extra-symbol legend from the relationship map — for looking things up without first having to select a specific distribution.