Beta Distribution Calculator
Calculate probabilities for the beta distribution with shape parameters α and β. Get P(X ≤ x), P(X ≥ x), or P(a ≤ X ≤ b), with interactive PDF/CDF graphs, shaded probability regions, step-by-step MathJax solutions, and distribution properties including mean, variance, mode, and skewness.
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About Beta Distribution Calculator
The Beta Distribution Calculator computes probabilities, visualizes the probability density function (PDF) and cumulative distribution function (CDF), and displays distribution properties for the beta distribution \(X \sim \text{Beta}(\alpha, \beta)\). Enter the shape parameters \(\alpha\) and \(\beta\) along with a value \(x \in [0, 1]\) to get \(P(X \leq x)\), \(P(X \geq x)\), or \(P(a \leq X \leq b)\), complete with step-by-step solutions, interactive graphs, and key statistics like the mean, variance, mode, and skewness.
What Is the Beta Distribution?
The beta distribution is a continuous probability distribution defined on the interval \([0, 1]\) with two positive shape parameters \(\alpha\) (alpha) and \(\beta\) (beta). Its probability density function (PDF) is:
$$f(x;\,\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad 0 \leq x \leq 1$$
where \(B(\alpha,\beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}\) is the beta function. The beta distribution is extremely versatile — by varying \(\alpha\) and \(\beta\), it can model uniform, bell-shaped, U-shaped, or J-shaped distributions, making it one of the most important distributions in probability and statistics.
Key Properties
Shape Gallery — How α and β Affect the Distribution
The beta distribution takes remarkably different shapes depending on its parameters:
Formulas
| Property | Formula | Description |
|---|---|---|
| \(f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\) | Probability density at x | |
| CDF | \(F(x) = I_x(\alpha,\beta)\) | Regularized incomplete beta function |
| Mean | \(\mu = \frac{\alpha}{\alpha+\beta}\) | Expected value |
| Variance | \(\sigma^2 = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\) | Spread of the distribution |
| Mode | \(\frac{\alpha-1}{\alpha+\beta-2}\) (if α, β > 1) | Most probable value |
| Skewness | \(\frac{2(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}\) | Asymmetry measure |
| Beta Function | \(B(\alpha,\beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\) | Normalization constant |
Bayesian Interpretation
The beta distribution is central to Bayesian statistics because it is the conjugate prior for the Bernoulli and Binomial distributions. If you have a prior belief about a probability \(p\) expressed as \(\text{Beta}(\alpha, \beta)\), and you observe \(s\) successes in \(n\) trials, then your updated (posterior) belief is:
$$p \mid \text{data} \sim \text{Beta}(\alpha + s, \; \beta + n - s)$$
This elegant update rule is why the beta distribution is the default choice for modeling uncertainty about probabilities. Common choices for priors include:
| Prior Name | Parameters | When to Use |
|---|---|---|
| Uniform (flat) | Beta(1, 1) | No prior information — all probabilities equally likely |
| Jeffreys prior | Beta(0.5, 0.5) | Non-informative prior with good mathematical properties |
| Haldane prior | Beta(0, 0) (improper) | Maximally non-informative — used in formal Bayesian analysis |
| Weak informative | Beta(2, 2) | Slight preference for values near 0.5 |
Real-World Applications
| Field | What X Models | Example |
|---|---|---|
| A/B Testing | Conversion rate probability | Estimating click-through rates for two website variants |
| Quality Control | Proportion of defective items | Modeling the defect rate of a manufacturing process |
| Sports Analytics | Win probability / batting average | Estimating a baseball player's true batting average |
| Insurance | Claim probability | Modeling the proportion of policyholders who file a claim |
| Genetics | Allele frequency | Modeling the frequency of a gene variant in a population |
| Machine Learning | Model confidence | Prior distribution for probability parameters in Bayesian classifiers |
Beta Distribution vs. Other Distributions
| Feature | Beta | Normal | Uniform |
|---|---|---|---|
| Support | [0, 1] | (−∞, +∞) | [a, b] |
| Parameters | α, β (shape) | μ, σ (location, scale) | a, b (endpoints) |
| Shape Flexibility | Very high (bell, U, J, flat) | Always bell-shaped | Always flat |
| Best For | Proportions, probabilities | Unbounded measurements | Equal-likelihood scenarios |
| Bayesian Use | Conjugate prior for Bernoulli | Conjugate prior for Normal (known σ) | Non-informative prior |
How to Use the Beta Distribution Calculator
- Enter the shape parameters α and β: Both must be positive numbers. α controls how much weight is near 1, and β controls weight near 0. For a symmetric distribution, set α = β.
- Select the probability type: Choose P(X ≤ x) for cumulative probability, P(X ≥ x) for survival probability, or P(a ≤ X ≤ b) for range probability.
- Enter the x value or range: Values must be between 0 and 1. For range probabilities, enter both lower bound a and upper bound b.
- Review the results: Examine the probability result, shape classification badge, interactive PDF and CDF graphs with shaded probability regions, distribution properties (mean, variance, mode), and the complete step-by-step solution.
FAQ
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"Beta Distribution Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: 2026-04-14
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