Why is its support the interval [0, 1]?

Because it is designed to model quantities that are proportions or probabilities: conversion rates, defect percentages, shares. Any bounded variable can be rescaled to [0, 1] to use it.

How should I interpret α and β?

They act as pseudo-counts: α plays the role of observed successes and β of failures. The mean is α/(α + β), and the larger α + β, the more concentrated the distribution.

Why is it central to Bayesian statistics?

Because it is the conjugate prior of the Bernoulli and binomial: if the prior on p is Beta(α, β) and you observe k successes in n trials, the posterior is Beta(α + k, β + n − k), with no numerical integration needed.

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Explanation

The beta distribution is a continuous distribution defined on the interval [0, 1], which makes it ideal for modeling proportions, rates and unknown probabilities. Its shape depends on two positive shape parameters α and β: when both are greater than 1 the distribution has a bell shape (symmetric if α = β, skewed otherwise); when α < 1 or β < 1 the density grows toward the ends of the interval. In Bayesian statistics it is the conjugate prior distribution of the binomial, which means that if you use a beta as your initial belief about a probability p, after observing binomial data the posterior distribution remains a beta with updated parameters. Its mean is α / (α + β) and its variance is αβ / [(α + β)²(α + β + 1)].

Formula

\( f(x;\alpha,\beta)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\quad 0\le x\le 1 \)

Parameters

α (alpha): shape parameter that controls behavior near x = 0. Must satisfy α > 0.
β (beta): shape parameter that controls behavior near x = 1. Must satisfy β > 0.
x: value of the proportion or probability, on the interval [0, 1].

Worked example

Situation: In a quality control trial, the proportion of conforming units \(X\) is modeled with a Beta distribution with parameters \(\alpha = 2\) and \(\beta = 5\). This choice reflects a moderate initial belief: most of the density lies at low values of \(X\), with mean \(\alpha/(\alpha+\beta) = 2/7 \approx 0.286\).

Question 1: What is the probability that the conformance rate is 0.3 or less, \(P(X \leq 0.3)\)?

Solution: We evaluate the CDF of Beta(2, 5) at \(x = 0.3\) using the regularized incomplete beta function \(I_{0.3}(2,5)\): \[ P(X \leq 0.3) = I_{0.3}(2,\,5) \approx 0.580 \] There is approximately a 58% probability that the conformance rate does not exceed 30%.

Question 2: What is the mode of the distribution, i.e., the most likely value?

Solution: For \(\alpha > 1\) and \(\beta > 1\), the mode of Beta(\(\alpha, \beta\)) is: \[ \text{mode} = \frac{\alpha - 1}{\alpha + \beta - 2} = \frac{2 - 1}{2 + 5 - 2} = \frac{1}{5} = 0.2 \] The most likely value of the conformance proportion under this model is 0.2 (20%).

Interpretation: The mode of 0.2 lies below the mean of 0.286, which reflects the right skew of this beta distribution. As more data are observed, the parameters \(\alpha\) and \(\beta\) are updated (Bayesian analysis) and the distribution narrows toward the true value of the rate.

How to interpret the result

The calculator offers three types of output. The PDF, \( f(x) \), gives the probability density at a point \( x \in [0, 1] \): it reflects how plausible that value of the proportion or probability is under the beta model with the chosen parameters \( \alpha \) and \( \beta \). A narrow, tall peak indicates strong concentration (high certainty), while a flat curve (as with \( \alpha = \beta = 1 \)) represents total uncertainty. In the chart, the green area corresponds to the probability that the variable falls within the selected range of the interval \([0, 1]\).

The CDF, \( P(X \leq x) \), gives the probability that the unknown proportion or rate is less than or equal to \( x \). In a Bayesian context, if \( X \) is the true conversion rate of a campaign, \( P(X \leq 0.10) = 0.03 \) means there is only a 3% probability that the true rate is below 10%, given what was observed. The right tail \( P(X > x) \) measures the probability that the proportion exceeds the threshold \( x \), and the difference between two CDFs gives the probability of a specific range.

The percentile or quantile result returns the value \( x \) such that \( P(X \leq x) = p \). The 2.5th and 97.5th percentiles form the 95% Bayesian credible interval: there is a 95% probability that the true parameter lies within that range. In A/B testing and Bayesian analysis, this interval is the Bayesian analogue of the frequentist confidence interval and is directly interpretable as a probabilistic statement about the unknown value.

Frequently asked questions

Why is its support the interval [0, 1]? Because it is designed to model quantities that are proportions or probabilities: conversion rates, defect percentages, shares. Any bounded variable can be rescaled to [0, 1] to use it.
How should I interpret α and β? They act as pseudo-counts: α plays the role of observed successes and β of failures. The mean is α/(α + β), and the larger α + β, the more concentrated the distribution.
Why is it central to Bayesian statistics? Because it is the conjugate prior of the Bernoulli and binomial: if the prior on p is Beta(α, β) and you observe k successes in n trials, the posterior is Beta(α + k, β + n − k), with no numerical integration needed.

Reference: Beta distribution — Wikipedia