The Distributions Lab
Decoding probabilistic uncertainty. Visual explorations of random variables and their distributions, grounded in intuition rather than pure computation.
Volume I — Binary & Discrete Uncertainty
Probability begins with the simplest question: did it happen or not? We start with the foundational atomic unit of uncertainty—the Bernoulli trial—and build upward into the discrete architectures that model our counting of events. These distributions form the basic grammar of statistical inference.
Bernoulli
The Atomic Unit of Uncertainty
A single trial with exactly two possible outcomes. The foundational building block for all discrete probability.
Binomial
In Development
The sum of multiple independent Bernoulli trials. Models the number of successes in a fixed number of binary experiments.
Geometric
In Development
Models the number of trials needed to get the first success in repeated Bernoulli trials.
Poisson
In Development
Expresses the probability of a given number of events occurring in a fixed interval of time or space.
Volume II — Continuous Worlds
Moving beyond simple counting, we enter the realm of the continuous. Here, variables take on infinite possible values, requiring new mathematical machinery. These are the models that describe everything from natural phenomena and measurement errors to the theoretical foundations of deep learning.
Uniform
In Development
The simplest continuous distribution, where all outcomes in an interval are equally likely.
Gaussian (Normal)
In Development
The ubiquitous bell curve. Arises naturally in many contexts due to the Central Limit Theorem.
Exponential
In Development
Describes the time between events in a Poisson point process, implying a memoryless property.
Chi-square
In Development
The distribution of a sum of the squares of k independent standard normal random variables. Crucial for hypothesis testing.
Beta
In Development
A family of continuous distributions defined on the interval [0, 1]. Often used as a prior distribution for probabilities.
Volume III — Learnable Probability Systems
Distributions are not just static descriptions; they are dynamic systems that can be updated as new data arrives. In this volume, we explore Bayesian priors, posteriors, and the distributions designed specifically to learn and adapt, serving as the core of modern machine learning.
Dirichlet
In Development
The multivariate generalization of the Beta distribution. Used to model probabilities over categorical variables.
Mixture Models
In Development
Probabilistic models for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs.
Concrete
In Development
A continuous relaxation of discrete random variables. Allows for backpropagation through categorical distributions using the Gumbel-Softmax trick.
Hard Concrete
In Development
An extension of the Concrete distribution that stretches the support to include exactly zero and one. Often used in sparse neural networks like L0 regularization.