Beta Distribution — Modeling Probabilities and Proportions
Foundations of Statistics
The Bayesian Workhorse for Probabilities
The beta distribution is the conjugate prior for binomial data, making it essential for Bayesian inference about proportions. Its flexibility on [0,1] makes it perfect for modeling uncertain probabilities and updating beliefs with data.
- A/B Testing — Updating conversion rate estimates as website test data accumulates
- Political Polling — Incorporating prior knowledge into election probability forecasts
- Quality Control — Modeling defect rates in manufacturing with Bayesian methods
The beta distribution turns prior knowledge into posterior certainty.
Core Concepts
The beta distribution is defined on and is the conjugate prior for the Bernoulli/Binomial likelihood in Bayesian inference. It provides a flexible family for modeling probabilities, proportions, and rates.
Interactive Visualization
The Beta Function
Mean, Variance, and Higher Moments
Symmetry and Shape
The Conjugate Prior Property
MGF and Moments
Connection to the F Distribution
Connection to the Binomial Distribution
Worked Example
Specific Applications
- Bayesian A/B testing — Prior/posterior on click-through rates, conversion rates.
- Bayesian statistics — Conjugate prior for Bernoulli, binomial, and negative binomial likelihoods.
- Modeling rates and proportions — Prevalence rates, completion rates, success probabilities.
- Project management —PERT distributions use beta to model task completion percentages.