Beta Distribution Calculator

Distributions and Statistical Models

Enter the shape parameters (α and β) to calculate the properties of the Beta distribution.

Examples

Click on an example to load its data into the calculator.

Right-Skewed Distribution

unimodal

A common scenario in Bayesian analysis where successes (α) are fewer than failures (β), pushing the peak towards 0.

α: 2, β: 5, x: 0.25

Left-Skewed Distribution

unimodal

Represents a scenario where successes (α) are more frequent than failures (β), pulling the distribution's peak towards 1.

α: 5, β: 2, x: 0.75

Symmetric Parabolic Distribution

symmetric

When α and β are equal and greater than 1, the distribution is symmetric and bell-shaped, centered at 0.5.

α: 2, β: 2, x: 0.5

U-Shaped Distribution

bimodal

When α and β are less than 1, the distribution is U-shaped, indicating that outcomes are more likely to be near 0 or 1.

α: 0.5, β: 0.5, x: 0.5

Other Titles
Understanding the Beta Distribution: A Comprehensive Guide
Explore the theory, application, and mathematics behind the Beta distribution, a versatile tool for modeling probabilities.

What is the Beta Distribution?

  • Core Concepts
  • Key Parameters
  • Versatile Shapes
The Beta distribution is a continuous probability distribution defined on the interval [0, 1]. It is parameterized by two positive shape parameters, denoted as alpha (α) and beta (β). Its primary use is to model the uncertainty about a probability. For instance, it can represent the probability of success in an experiment, like the click-through rate of an ad or the conversion rate of a website.
The Role of Alpha (α) and Beta (β)
The parameters α and β are the 'shape' parameters that define the form of the distribution. Intuitively, they can be thought of as counts of 'successes' (α) and 'failures' (β). If α is greater than β, the distribution's mass is concentrated towards 1 (high probability of success). Conversely, if β is greater than α, the mass is concentrated towards 0 (low probability of success). When α and β are equal, the distribution is symmetric around 0.5.

Step-by-Step Guide to Using the Beta Distribution Calculator

  • Entering Parameters
  • Interpreting the Results
  • Using Examples
This calculator simplifies the process of working with the Beta distribution. Follow these steps:
• Enter the Alpha (α) value: This must be a positive number representing the count of successes or evidence for a higher probability.
• Enter the Beta (β) value: This must be a positive number representing the count of failures or evidence for a lower probability.
• Enter the x Value (optional): Provide a value between 0 and 1 to calculate the Probability Density Function (PDF) and Cumulative Distribution Function (CDF) at that specific point.
• Click 'Calculate': The tool will compute the mean, variance, standard deviation, mode, PDF, and CDF.
Understanding the Outputs
The results give a full picture of the distribution. The mean gives the average expected probability, while the variance shows its spread. The PDF tells you the likelihood of a specific probability 'x', and the CDF tells you the total probability of getting a value up to 'x'.

Real-World Applications of the Beta Distribution

  • Bayesian Inference
  • Project Management
  • A/B Testing
Bayesian Inference and Prior Distributions
The Beta distribution is famously used in Bayesian statistics as a conjugate prior for the binomial distribution. This means if you have a prior belief about a probability (modeled as a Beta distribution) and you observe new data (from a binomial experiment), your updated belief (the posterior) is also a Beta distribution. This makes updating beliefs mathematically convenient.
Task Duration Modeling in PERT
In project management, the PERT (Program Evaluation and Review Technique) uses the Beta distribution to model the time a task might take. By estimating optimistic, pessimistic, and most likely completion times, a Beta distribution can be fitted to estimate the expected time and risk.
A/B Testing Analysis
In marketing and product development, A/B testing is used to compare two versions of a webpage or app. The conversion rate for each version can be modeled as a Beta distribution. By comparing the two distributions, one can determine the probability that version A is better than version B.

Common Misconceptions and Correct Methods

  • Confusing with Normal Distribution
  • Interpreting the Mode
  • Choosing Priors
One common mistake is to assume all probability distributions are bell-shaped like the Normal distribution. The Beta distribution is highly flexible and can be U-shaped, J-shaped, or uniform, not just bell-shaped.
When is the Mode Meaningful?
The formula for the mode, (α - 1) / (α + β - 2), is only valid when both α and β are greater than 1. If either is less than or equal to 1, the distribution's peak is at an endpoint (0 or 1) or the distribution is U-shaped, in which case the mode as a single peak is not well-defined in the same way.
Informative vs. Uninformative Priors
In Bayesian analysis, choosing priors is crucial. A Beta(1, 1) distribution is a uniform distribution, representing no prior knowledge (an uninformative prior). Using Beta(0, 0) is sometimes suggested but is an improper prior. An informative prior, like Beta(10, 2), would strongly suggest the probability of success is high.

Mathematical Derivation and Formulas

  • Probability Density Function (PDF)
  • Cumulative Distribution Function (CDF)
  • Key Statistical Measures
The mathematical foundation of the Beta distribution lies in the Beta function, B(α, β).
The PDF Formula
The PDF is given by: f(x; α, β) = (x^(α-1) * (1-x)^(β-1)) / B(α, β), where B(α, β) = Γ(α)Γ(β) / Γ(α+β) is the Beta function, and Γ is the Gamma function. This formula ensures the total area under the curve is 1.
The CDF Formula
The CDF is the regularized incomplete beta function, I_x(α, β), which represents the integral of the PDF from 0 to x. It does not have a simple closed-form expression and is typically computed numerically.
Formulas for Mean, Variance, and Mode
• Mean (E[X]): α / (α + β)
• Variance (Var(X)): (αβ) / ((α + β)²(α + β + 1))
• Mode: (α - 1) / (α + β - 2) (for α, β > 1)