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Beta Distribution Calculator

Free Beta distribution calculator. Compute PDF, CDF, mean, mode, variance, P(a≤X≤b), percentiles. Ba

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Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

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STATISTICSDistributions

Beta Distribution — PDF, CDF, Mean, Mode for Beta(α,β)

Bayesian priors and A/B testing. Conjugate prior for proportions. P(a≤X≤b), percentiles, and step-by-step breakdown.

Binomial →Bayes Theorem →

Real-World Scenarios — Click to Load

Inputs

Beta PDF — Shaded P(a≤X≤b)

beta_results.sh
CALCULATED
$ beta_dist --alpha=2 --beta=5 --a=0 --b=1
Mean
0.2857
Mode
0.2000
Variance
0.025510
P(a≤X≤b)
100.0000%
25th %ile
0.1610
50th %ile
0.2642
75th %ile
0.3891
95th %ile
0.5822
Share:
Beta Distribution
Beta(2, 5)
P(0≤X≤1) = 100.00%
Mean = 0.286SD = 0.160Skewness = 0.60
numbervibe.com/calculators/statistics/beta-distribution-calculator

CDF F(x)

Reference Shapes: Beta(2,5), Beta(5,5), Beta(8,2)

Calculation Breakdown

COMPUTATION
Mean
0.2857
α/(α+β) = 2/(2+5)
Mode
0.2000
ext{Mode} = rac{\text{alpha} -1}{\text{alpha} +\text{beta} -2} ext{when} \text{alpha} ,\text{beta} > 1
MOMENTS
Variance
0.025510
αβ/[(α+β)²(α+β+1)]
P(a≤X≤b)
100.0000%
I_b(\text{alpha} ,\text{beta} ) - I_a(\text{alpha} ,\text{beta} )
POINT EVALUATION
PDF at x
2.160900
f(x) = x^(\text{alpha} -1)(1-x)^(\text{beta} -1)/B(\text{alpha} ,\text{beta} )
CDF at x
0.580471
I_x(\text{alpha} ,\text{beta} )

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

  • • The Beta distribution Beta(α, β) models probabilities on [0,1] — ideal for proportions and Bayesian priors
  • • Mean = α/(α+β), Variance = αβ/[(α+β)²(α+β+1)], Mode = (α-1)/(α+β-2) when α,β > 1
  • • Conjugate prior for the binomial: Beta is the natural prior for success probability p
  • • Bayesian updating: Prior Beta(α,β) + k successes in n trials → Posterior Beta(α+k, β+n-k)
  • • A/B testing: Use Beta to model conversion rates and compute P(A > B)

Did You Know?

📊The Beta(1,1) is the uniform distribution — the uninformative prior for a probability.Source: Bayesian prior
🧪In A/B testing, Beta distributions model conversion rates; P(variant A > B) is computed via Monte Carlo or closed form.Source: A/B testing
📐Project management: Beta is used in PERT for task duration estimates (optimistic, most likely, pessimistic).Source: PERT
🎯Strong prior Beta(50,50) concentrates mass near 0.5 — use when you have strong prior belief.Source: Gelman BDA
📈α < β gives right skew (high density near 0); α > β gives left skew (high density near 1).Source: Shape parameters
🔬Gelman et al. BDA recommends Beta(2,2) or Beta(1,1) as default priors for binomial proportions.Source: BDA textbook

How It Works

1. PDF Formula

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.

2. CDF — Regularized Incomplete Beta

P(X ≤ x) = I_x(α, β) — the regularized incomplete beta function. P(a ≤ X ≤ b) = I_b − I_a.

3. Bayesian Conjugate Prior

Prior Beta(α,β) + binomial likelihood(k|n,p) → Posterior Beta(α+k, β+n-k). No MCMC needed.

4. A/B Testing

Model each variant's conversion rate as Beta. Sample from both posteriors and compare P(A > B).

5. Percentiles

The p-th percentile is x such that I_x(α,β) = p. Found via binary search on the CDF.

Expert Tips

Uninformative Prior

Use Beta(1,1) when you have no prior information about the proportion.

Jeffreys Prior

Beta(0.5, 0.5) is the Jeffreys prior for the binomial — invariant to reparameterization.

Mode Edge Cases

Mode exists only when α,β > 1. For α=β=1 (uniform), every point is a mode.

PERT Estimates

Use Beta with α,β derived from min/mode/max for project task duration modeling.

Why Use This Calculator vs Other Tools?

FeatureThis CalculatorExcelRSciPy
PDF + CDF + Percentiles✅⚠️ Multiple functions⚠️ Multiple functions✅
Shaded region P(a≤X≤b)✅❌❌⚠️ Manual
7 example presets✅❌❌❌
Bayesian/A/B context✅❌❌❌
Educational content✅❌❌❌

Frequently Asked Questions

What is the Beta distribution used for?

Modeling proportions and probabilities on [0,1]. Common in Bayesian statistics (conjugate prior for binomial), A/B testing (conversion rates), and project management (PERT).

What does conjugate prior mean?

When the prior and likelihood are from the same family, the posterior is also in that family. Beta + Binomial → Beta. This allows closed-form Bayesian updating.

How do I choose α and β for a prior?

Beta(1,1) is uninformative. For a prior mean μ and effective sample size n0, use α = μ·n0, β = (1-μ)·n0. Beta(2,2) is a weak prior centered at 0.5.

When does the mode exist?

Mode = (α-1)/(α+β-2) exists when α > 1 and β > 1. For α=β=1 (uniform), there is no unique mode.

How is the Beta related to the binomial?

The Beta is the conjugate prior for the binomial success probability p. After observing k successes in n trials, posterior is Beta(α+k, β+n-k).

What is PERT and how does it use Beta?

PERT uses a Beta-like distribution for task duration: mean ≈ (a+4m+b)/6 where a=min, m=mode, b=max. The Beta can model this with appropriate α,β.

Beta Distribution by the Numbers

[0,1]
Support
α, β
2 Parameters
Conjugate
Binomial Prior
I_x(α,β)
CDF Formula

Disclaimer: This calculator uses numerical integration for the regularized incomplete beta function. Results are accurate for typical parameter ranges (α, β in 0.01–100). For critical applications, verify against established statistical software (R, SciPy, MATLAB).

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