Beta Distribution Calculator
Enter shape parameters α > 0 and β > 0 directly. Optionally specify an x value for PDF/CDF, or a probability p for the quantile (inverse CDF).
Start with a Beta prior (αprior, βprior), then enter observed successes k and failures f. The posterior is Beta(α + k, β + f).
Convert a 3-point PERT estimate (Minimum, Most Likely, Maximum) into Beta distribution shape parameters for project scheduling and risk analysis.
Run a calculation in any of the three tabs first, then return here to see the full step-by-step solution.
No data yet — enter values in Standard, Bayesian Update, or PERT first.
What Is the Beta Distribution?
The Beta distribution is a family of continuous probability distributions defined on the closed interval [0, 1], parameterized by two positive shape parameters α (alpha) and β (beta). Because its support is bounded between 0 and 1, it naturally models quantities that are themselves bounded proportions: conversion rates, defect rates, survey response fractions, or any other value that cannot fall below zero or exceed one.
The two shape parameters give the Beta family its flexibility. By tuning α and β, you can produce distributions that are uniform, bell-shaped, J-shaped, U-shaped, or heavily skewed in either direction. This makes the Beta distribution the go-to choice wherever you need to model uncertainty about an unknown probability or proportion — particularly within the Bayesian statistical framework, where it serves as the conjugate prior for binomial data.
Probability Density Function and CDF Formulas
The Beta PDF tells you how densely probability is concentrated at any point x in [0, 1]. The CDF tells you the total probability accumulated up to a given point. Both rely on the Beta function B(α, β) as a normalizing constant.
Probability Density Function (PDF)
f(x; α, β) = xα−1(1−x)β−1 / B(α, β)
Where B(α, β) = Γ(α)Γ(β) / Γ(α+β)
and x ∈ [0, 1], α > 0, β > 0
Cumulative Distribution Function (CDF)
F(x; α, β) = Ix(α, β)
Where Ix(α, β) is the regularized
incomplete Beta function:
= B(x; α, β) / B(α, β)
Summary Statistics
Mean: E[X] = α / (α + β)
Variance: αβ / [(α+β)²(α+β+1)]
Mode: (α−1) / (α+β−2) when α,β > 1
Skewness: 2(β−α)√(α+β+1) / [(α+β+2)√(αβ)]
Mode — All Cases
α > 1, β > 1: (α−1) / (α+β−2)
α = 1, β > 1: 0
α > 1, β = 1: 1
α < 1, β < 1: bimodal {0, 1}
α = 1, β = 1: uniform on [0,1]
The probability density function integrates to 1 over [0, 1] by construction, guaranteed by the normalizing Beta function B(α, β). Computing the CDF numerically requires evaluating the regularized incomplete Beta function, which is what this calculator handles automatically.
Shape Parameter Behavior: What α and β Control
The shape parameters do not merely shift the distribution — they change its fundamental character. The table below maps parameter conditions to curve types, so you can choose α and β to match the real-world behavior you want to model.
| Parameter Condition | Curve Type | Visual Shape | Primary Use Case |
|---|---|---|---|
| α = 1, β = 1 | Uniform | Flat line at f(x) = 1 | Uninformative Bayesian prior; complete uncertainty |
| α = β > 1 | Symmetric Bell | Unimodal, peaked at x = 0.5 | Coin-flipping models, balanced probability estimation |
| α < β, both > 1 | Right-Skewed | Peak shifted toward x = 0 | Low conversion rates, rare-event modeling |
| α > β, both > 1 | Left-Skewed | Peak shifted toward x = 1 | High pass rates, customer retention analysis |
| α < 1, β < 1 | U-Shaped | Spikes at both boundaries 0 and 1 | Polarized outcomes, binary state switching |
| α = 1, β > 1 | Decreasing | Peak at x = 0, decays monotonically | Time-to-failure ratios, rare-event rates |
Worked Example: E-Commerce Conversion Rate
Here is how a product manager might apply the Beta distribution when analyzing a new checkout flow, starting with a prior belief and working through each calculation.
Step 1 — Mean: E[X] = 2 / (2 + 5) = 2/7 ≈ 0.2857 (28.57%). This is the best single-number estimate of the conversion rate under this prior.
Step 2 — Variance: Var(X) = (2 × 5) / [(7)² × (7 + 1)] = 10 / 392 ≈ 0.02551. The standard deviation is √0.02551 ≈ 0.160.
Step 3 — Mode: Mode = (2 − 1) / (2 + 5 − 2) = 1/5 = 0.20 (20%). The most probable single rate is 20%.
Step 4 — CDF at x = 0.30: Evaluating the regularized incomplete Beta function gives I0.30(2, 5) ≈ 0.5798 (57.98%). There is roughly a 58% chance the true conversion rate is at or below 30%.
Bayesian A/B Testing with the Beta Conjugate Prior
The Beta distribution's most powerful property is its conjugacy with the Binomial likelihood. If your prior belief is Beta(α, β) and you then observe k successes and f failures, the posterior updates to Beta(α + k, β + f) without any integration. This makes sequential Bayesian updating practical.
In A/B testing, a common approach is to start each variant with a Beta(1, 1) uninformative prior, collect data, and watch the posterior concentrate around the true conversion rate as sample size grows. The probability that variant A beats variant B is then computed by numerical integration of the two posterior Beta distributions — something this calculator handles in the Bayesian Update tab.
Updating in Practice
Suppose you start with Beta(1, 1) (uniform, no prior knowledge) and you observe 30 conversions from 100 visitors. The 70 non-converting visitors are failures, so the posterior becomes Beta(1 + 30, 1 + 70) = Beta(31, 71). The posterior mean is 31 / (31 + 71) = 31/102 ≈ 0.304 (30.4%), and the posterior is already reasonably tight. Running the Bayesian Update tab above replicates this instantly.
Code Implementation
The three code blocks below show how to replicate the calculator's outputs in Python, R, and spreadsheet software.
Python (scipy.stats)
from scipy import stats
alpha, beta = 2, 5
# Summary statistics
mean = stats.beta.mean(alpha, beta) # 0.2857
var = stats.beta.var(alpha, beta) # 0.02551
# PDF at x = 0.30
pdf_val = stats.beta.pdf(0.30, alpha, beta)
# CDF: P(X <= 0.30)
cdf_val = stats.beta.cdf(0.30, alpha, beta) # ≈ 0.5798
# Quantile (Inverse CDF): find x at 95th percentile
q_95 = stats.beta.ppf(0.95, alpha, beta)
R Language
alpha <- 2
beta <- 5
# PDF at x = 0.30
pdf_val <- dbeta(0.30, shape1 = alpha, shape2 = beta)
# CDF at x = 0.30
cdf_val <- pbeta(0.30, shape1 = alpha, shape2 = beta)
# Inverse CDF for p = 0.95
q_val <- qbeta(0.95, shape1 = alpha, shape2 = beta)
Microsoft Excel / Google Sheets
=BETA.DIST(0.30, 2, 5, FALSE)
=BETA.DIST(0.30, 2, 5, TRUE)
=BETA.INV(0.95, 2, 5)
Related Calculators and Guides
The Beta distribution connects closely to several other probability concepts. The guides and tools below build on what you have learned here.
Frequently Asked Questions
The Beta distribution models continuous variables constrained to [0, 1] — probabilities, proportions, success rates, and conversion metrics. In Bayesian statistics, it is the standard conjugate prior for Binomial experiments because updating it with observed data produces another Beta distribution, keeping computation tractable. Outside Bayesian work, it appears in project scheduling (PERT), quality control (batch contamination rate estimation), and anywhere you need to quantify uncertainty about an unknown proportion.
A prior distribution is conjugate for a given likelihood when the posterior falls in the same distribution family as the prior. For a Binomial likelihood with unknown success probability p, if the prior is Beta(α, β), the posterior after observing k successes and f failures is Beta(α + k, β + f). The update is just parameter addition — no integrals required. This mathematical convenience is why Bayesian A/B testing frameworks almost universally model conversion rates as Beta-distributed.
The mean is E[X] = α / (α + β). For Beta(2, 5) the mean is 2 / 7 ≈ 0.2857. Intuitively, α counts the “successes” and β the “failures” in a pseudo-sample, so the mean is simply the proportion of successes. As α grows relative to β, the mean rises toward 1; as β grows relative to α, it falls toward 0.
Beta(1, 1) is identical to the standard Uniform distribution on [0, 1], where f(x) = 1 for every x. It represents total ignorance about the unknown probability — every value in [0, 1] is equally plausible before data. This is the classic uninformative (or “flat”) prior in Bayesian analysis, though some statisticians prefer the Jeffreys prior Beta(0.5, 0.5) instead, which is U-shaped rather than flat.
Yes, both parameters can be any positive real number, including values between 0 and 1. When both are less than 1, the PDF is U-shaped: density is infinite at the boundaries 0 and 1, and lowest in the middle. This models polarized outcomes where extreme probabilities are more credible than moderate ones. When only one parameter is below 1, the distribution is J-shaped, increasing steeply toward one boundary.
PERT (Program Evaluation and Review Technique) fits a four-parameter Beta distribution to a 3-point estimate: minimum a, most likely m, and maximum b. The shape parameters are derived as α = 1 + 4(m − a)/(b − a) and β = 1 + 4(b − m)/(b − a), and the PERT mean is (a + 4m + b) / 6. The standard deviation is (b − a) / 6. This bounded approach is preferred over the normal distribution for task durations because it respects the hard lower and upper limits.