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Beta Distribution

Created:October 5, 2024
Last Updated:March 26, 2025

This calculator allows you to calculate the probability of a beta distribution falling within a specified range or being greater than or less than a certain value. It also provides a chart and accompanying R code that you can run locally. Simply input the alpha (α) and beta (β) parameters, the comparison type, and the value(s) to compare against to get started.

Calculator

Parameters

Distribution Chart

Click Calculate to view the distribution chart

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Beta Distribution: Definition, Formula, and Applications

Beta Distribution

Definition: The beta distribution is a family of continuous probability distributions defined on the interval [0, 1]. It is characterized by two positive shape parameters, α (alpha) and β (beta), which determine its shape and properties.

Formula:The probability density function (PDF) is given by:f(x;α,β)=Γ(α+β)Γ(α)Γ(β)xα−1(1−x)β−1,0≤x≤1f(x; \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}, \quad 0 \leq x \leq 1where:Γ(z)=∫0∞tz−1e−tdt\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} dt

Where:

  • α>0\alpha > 0 is the first shape parameter
  • β>0\beta > 0 is the second shape parameter
  • Γ(z)\Gamma(z) is the gamma function

Properties

Key Statistics:

  • Mean: E(X)=αα+βE(X) = \frac{\alpha}{\alpha + \beta}
  • Variance: Var(X)=αβ(α+β)2(α+β+1)\text{Var}(X) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}
  • Mode:
    • α−1α+β−2\frac{\alpha - 1}{\alpha + \beta - 2} for α,β>1\alpha, \beta > 1
    • 0 when α=1,β>1\alpha = 1, \beta > 1
    • 1 when α>1,β=1\alpha > 1, \beta = 1
    • Undefined when α,β<1\alpha, \beta < 1

Special Cases:

  • α=β=1\alpha = \beta = 1: Uniform distribution on [0,1]
  • α=β\alpha = \beta: Symmetric distribution around 0.5
  • α,β<1\alpha, \beta < 1: U-shaped distribution
  • α,β>1\alpha, \beta > 1: Unimodal distribution

How to Calculate Beta Distribution in R?

R
library(tidyverse)

alpha <- 2
beta <- 5

# Calculate probability between two values
x1 <- 0.2
x2 <- 0.6
prob <- pbeta(x2, shape1 = alpha, shape2 = beta) - pbeta(x1, shape1 = alpha, shape2 = beta)
print(str_glue("P({x1} < X < {x2}) = {round(prob, 4)}"))

# P(0.2 < X < 0.6) = 0.6144

How to Calculate Beta Distribution in Python?

Python
import numpy as np
import pandas as pd
from scipy import stats

alpha = 2
beta = 5

# Calculate probability between two values
x1, x2 = 0.2, 0.6
prob = stats.beta.cdf(x2, alpha, beta) - stats.beta.cdf(x1, alpha, beta)
print(f"P({x1} < X < {x2}) = {prob:.4f}")

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