Calculators
Beta Distribution Calculator
Calculate Beta distribution probabilities, quantiles, density, mean, variance, and mode for shape parameters α and β locally in your browser.
Model probabilities and proportions with the Beta distribution.
The Beta distribution is a continuous distribution on [0, 1] defined by two positive shape parameters α and β. It is the conjugate prior for the binomial distribution in Bayesian statistics and models uncertain probabilities. Enter α and β from 0.01 to 10000, and either x in [0, 1] for probabilities (density, CDF, upper tail) or cumulative probability p in (0, 1) for quantiles. The calculator returns mean α/(α+β), variance αβ/((α+β)²(α+β+1)), standard deviation, and mode (α−1)/(α+β−2) when α, β > 1. All calculations run locally in your browser; your values are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What is a Beta distribution?
The Beta distribution is a continuous probability distribution defined on the interval [0, 1] with two positive shape parameters α (alpha) and β (beta). It is commonly used to model probabilities, proportions, and rates. In Bayesian statistics, it is the conjugate prior for the binomial distribution.
What formulas does this calculator use?
The calculator uses: Mean = α/(α+β), Variance = αβ/((α+β)²(α+β+1)), Standard deviation = √Variance, Mode = (α−1)/(α+β−2) for α, β > 1. The density is f(x) = x^(α−1)(1−x)^(β−1)/B(α,β) where B is the Beta function. The CDF uses the regularized incomplete beta function I_x(α,β). Quantiles use bisection of the regularized incomplete beta function. Supported shape parameters range from 0.01 to 10000; extreme results may round to 0 or 1. Boundary modes and the uniform case are identified separately.
When should I use a Beta distribution?
Use it when modeling a quantity that is naturally bounded between 0 and 1, such as a probability, proportion, or rate. It is the conjugate prior for the binomial likelihood in Bayesian inference. It is also used in project management (PERT) for task duration estimates and in order statistics.
What is the difference between probability mode and quantile mode?
In probability mode, you enter a value x in [0, 1] and get the density f(x), cumulative probability P(X ≤ x), and upper tail P(X > x). In quantile mode, you enter a cumulative probability p in (0, 1) and get the value x such that P(X ≤ x) = p.
Do my values stay private?
Yes. All calculations run locally in your browser. CodeASystem does not upload or store the values you enter.