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

🔗

Bayesian Conjugate

Conjugate prior for Bernoulli, Binomial, and Geometric distributions in Bayesian inference.

🔄

Shape Versatility

Can model uniform, U-shaped, J-shaped, and bell-shaped distributions depending on α and β.

📏

Bounded Support

Defined on [0, 1], making it ideal for modeling probabilities, proportions, and rates.

⚖

Symmetry

When α = β, the distribution is symmetric about 0.5. Otherwise it is skewed.

Shape Gallery — How α and β Affect the Distribution

The beta distribution takes remarkably different shapes depending on its parameters:

Uniform

α = 1, β = 1

Symmetric Bell

α = 5, β = 5

J-Shaped (left)

α = 0.5, β = 2

U-Shaped

α = 0.5, β = 0.5

Formulas

Property	Formula	Description
PDF	$f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}$	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

What is the beta distribution?

The beta distribution is a continuous probability distribution defined on the interval [0, 1] with two shape parameters alpha and beta. It is extremely versatile and can model proportions, probabilities, and rates. Depending on the parameter values, it can take many shapes including bell-shaped, U-shaped, J-shaped, or uniform.

What are the shape parameters alpha and beta?

Alpha (α) and beta (β) are positive real numbers that control the shape of the distribution. When alpha equals beta, the distribution is symmetric around 0.5. When alpha is greater than beta, the distribution is skewed toward 1 (left-skewed). When alpha is less than beta, it is skewed toward 0 (right-skewed). When both are less than 1, the distribution becomes U-shaped with peaks at both endpoints.

How is the beta distribution used in Bayesian statistics?

The beta distribution is the conjugate prior for the Bernoulli and binomial distributions. In Bayesian inference, if you start with a Beta(α, β) prior for a probability parameter and observe s successes in n trials, the posterior distribution is Beta(α + s, β + n − s). This makes it the standard choice for modeling uncertainty about probabilities, conversion rates, proportions, and similar quantities bounded between 0 and 1.

What is the difference between the beta distribution and the normal distribution?

The beta distribution is bounded on [0, 1] while the normal distribution spans all real numbers from negative infinity to positive infinity. The beta distribution is far more flexible in shape — it can be U-shaped, J-shaped, bell-shaped, or flat — while the normal distribution is always bell-shaped. The beta distribution is the natural choice for modeling proportions and probabilities, while the normal distribution is best for unbounded continuous measurements.

How do I calculate the CDF of the beta distribution?

The CDF of the beta distribution is the regularized incomplete beta function I_x(α, β), defined as the integral from 0 to x of t^(α−1)(1−t)^(β−1) dt, divided by the beta function B(α, β). This integral has no closed-form solution for general α and β, so it is computed numerically using continued fraction algorithms or series expansions. This calculator handles the computation automatically.

Reference this content, page, or tool as:

"Beta Distribution Calculator" at https://MiniWebtool.com/beta-distribution-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-14

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About Beta Distribution Calculator

What Is the Beta Distribution?

Key Properties

Shape Gallery — How α and β Affect the Distribution

Formulas

Bayesian Interpretation

Real-World Applications

Beta Distribution vs. Other Distributions

How to Use the Beta Distribution Calculator

FAQ

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Statistics And Data Analysis:

Top & Updated:

Property	Formula	Description
PDF	\(f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\)	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