What is a Beta distribution?
The Beta distribution is a continuous distribution over (0,1). It’s commonly used for probabilities, rates, and proportions.
- α=β=1: uniform.
- α<1 and β<1: U-shaped (mass near 0 and 1).
- α>β: skewed toward 1. α<β: skewed toward 0.
- Large α and β: concentrated around the mean.
PDF: f(x)=x^(α-1)(1-x)^(β-1)/B(α,β). Mean: α/(α+β). Variance: αβ/[(α+β)^2(α+β+1)].
You don’t need to enter personal information to use this tool.
Presets
Quickly set common shapes (you can tweak values after applying).
Generator
Set α/β, sample size, bins, and RNG. Then generate samples and export results.
Sample stats
Samples (first 20)
How to use this tool
Use this page for probabilities, rates, and proportions that must stay between 0 and 1.
Use in 3 steps
- Set
αandβbased on the shape you expect near 0, near 1, or around the center. - Generate a sample and compare the histogram with the theoretical mean and variance.
- Adjust one parameter at a time to see whether the change affects left-tail mass, right-tail mass, or overall concentration.
How to read the result
The chart shows how mass is distributed between 0 and 1. Large α and β tighten the sample around the middle, while values below 1 can push density toward the edges.
Boundary checks
- If
α<1, density can rise sharply near 0. - If
β<1, density can rise sharply near 1. - Remember that Beta is the
K=2special case of Dirichlet when you compare tools.
Frequently asked questions
What do α and β control?
They control where the mass sits between 0 and 1 and how concentrated the distribution is around the center or the edges.
Why do values pile up near 0 or 1?
When α or β drops below 1, the density can spike near that boundary instead of staying smooth in the middle.
How is this related to Dirichlet?
A Beta distribution is the two-component Dirichlet case viewed through one component.
Why might two tools disagree?
Different parameter meanings, rounding, random seeds, or sample sizes can change the visible output.
What should I do first on this page?
Begin with a symmetric case such as α=β, then move only one side to see which boundary gets more weight.