Beta Distribution Calculator | PDF, CDF, Mean & Bayesian Inference
Compute the probability density function (PDF), cumulative distribution function (CDF), mean, variance, mode, and skewness of the Beta(α, β) distribution. Evaluates P(X ≤ x) and P(a ≤ X ≤ b) numerically, with applications in Bayesian inference, proportion modeling, and project completion time estimation.
Distribution Parameters
What Is the Beta Distribution Calculator | PDF, CDF, Mean & Bayesian Inference?
The Beta distribution is a continuous probability distribution defined on the interval [0,1], making it ideal for modeling proportions, probabilities, and fractions. Its two positive shape parameters α and β control the shape: both >1 gives a unimodal bell, both <1 gives a U-shaped bimodal distribution, and α=β=1 gives the uniform distribution. It is the conjugate prior for Bernoulli and Binomial likelihoods in Bayesian inference.
Formula
f(x; α, β) = x^(α−1)(1−x)^(β−1) / B(α,β) for x ∈ [0,1], where B(α,β) = Γ(α)Γ(β)/Γ(α+β)
How to Use
- 1
Enter α (alpha) and β (beta), both must be strictly positive real numbers. Try the preset buttons to load common configurations like Uniform or Bayesian coin flip.
- 2
Enter the evaluation point x in [0, 1]. The calculator will compute the PDF value f(x) and cumulative probability P(X ≤ x) at this point.
- 3
Enter interval bounds a and b to compute P(a ≤ X ≤ b). Leave them at default (0 and 1) to compute the total probability (always 1).
- 4
Click Calculate to see PDF, CDF, interval probability, and distribution statistics.
- 5
Read the statistics cards for mean α/(α+β), variance, mode (if α,β > 1), and skewness.
- 6
Review the PDF/CDF table showing values at x = 0, 0.1, 0.2, … 1.0 to understand the shape of the distribution.
- 7
For Bayesian updating: start with Beta(1,1) uniform prior, observe k successes in n trials, and update to Beta(1+k, 1+n−k).
Enter shape parameters α and β, an evaluation point x, and optionally an interval [a, b], then click Calculate.
Example Calculation
Bayesian coin-flip posterior: start with a Beta(1,1) uniform prior (no information). After observing 7 heads in 10 flips, the posterior is Beta(8,4). The mean is 8/12 ≈ 0.667, mode is 7/10 = 0.7. P(p > 0.5) = 1 − CDF(0.5; 8,4) ≈ 0.855, so there is about an 85.5% posterior probability the coin is biased toward heads.
Understanding Beta Distribution | PDF, CDF, Mean & Bayesian Inference
Special Cases of the Beta Distribution
By varying α and β you obtain many well-known distributions as special or limiting cases.
| Beta(α, β) | Common name | Shape | Mean | Mode |
|---|---|---|---|---|
| Beta(1, 1) | Uniform U[0,1] | Flat / constant | 0.5 | Any point |
| Beta(2, 2) | Symmetric bell | Unimodal, symmetric | 0.5 | 0.5 |
| Beta(0.5, 0.5) | Arcsine distribution | U-shaped (bimodal) | 0.5 | 0 and 1 |
| Beta(α, 1) | Power distribution | Monotone increasing for α>1 | α/(α+1) | 1 |
| Beta(1, β) | Reflected power | Monotone decreasing for β>1 | 1/(1+β) | 0 |
| Beta(2, 5) | Right-skewed | Peak near 0.2–0.3 | 2/7 ≈ 0.286 | 1/6 ≈ 0.167 |
| Beta(5, 2) | Left-skewed | Peak near 0.7–0.8 | 5/7 ≈ 0.714 | 5/6 ≈ 0.833 |
| Beta(α→∞, β→∞) | Approx. Normal | Bell curve on [0,1] | α/(α+β) | (α−1)/(α+β−2) |
Bayesian Inference with the Beta Distribution
The Beta distribution is the conjugate prior for the Bernoulli and Binomial likelihoods. This means the posterior is also Beta, making analytical updating trivial.
| Component | Distribution | Meaning |
|---|---|---|
| Prior | Beta(α₀, β₀) | Your belief about success probability p before seeing data |
| Likelihood | Binomial(n, p) | n trials with k successes observed |
| Posterior | Beta(α₀+k, β₀+n−k) | Updated belief after observing k successes in n trials |
| Example | Prior Beta(1,1) + 7 heads in 10 flips | Posterior = Beta(8,4), mean = 8/12 ≈ 0.667 |
| Credible interval | P(a ≤ p ≤ b | data) | Use CDF of posterior Beta to compute probability intervals |
Applications of the Beta Distribution
- ▸Bayesian statistics: Modeling uncertainty about a probability parameter p ∈ [0,1], such as a conversion rate, click-through rate, or disease prevalence.
- ▸Project management (PERT): The Program Evaluation and Review Technique uses Beta(2,2) or similar to model task duration as a fraction of its range, feeding into expected completion time calculations.
- ▸A/B testing: Updating beliefs about which variant has a higher conversion rate using Bayesian updates, computing P(variant A > variant B) from posterior betas.
- ▸Reliability engineering: Modeling the proportion of defective items in a batch or the probability a component survives a given stress level.
- ▸Machine learning: Latent Dirichlet Allocation (LDA) for topic modeling uses symmetric Beta priors over topic-word and document-topic distributions.
- ▸Finance: Modeling loss-given-default (LGD) rates in credit risk, which are bounded between 0% and 100%.
Frequently Asked Questions
Why is the Beta distribution defined only on [0,1]?
The Beta distribution's support [0,1] makes it natural for modeling quantities that are themselves proportions or probabilities. The PDF is zero outside [0,1] by definition. If you need to model a proportion on a different range [a, b], use a shifted and scaled Beta distribution: Y = a + (b−a)X where X ~ Beta(α,β).
What happens when α or β is less than 1?
When α < 1, the PDF goes to infinity as x → 0, creating a spike at the left boundary. When β < 1, there is a spike at x = 1. When both α < 1 and β < 1 (like the Arcsine distribution Beta(0.5, 0.5)), the distribution is U-shaped with spikes at both endpoints, concentrating probability near the boundaries. The mode formula (α−1)/(α+β−2) is undefined for α ≤ 1 or β ≤ 1.
How is the regularized incomplete beta function computed?
The CDF of the Beta distribution is the regularized incomplete beta function I_x(α, β) = B(x; α,β)/B(α,β). It is computed using Lentz's continued fraction algorithm, which converges rapidly. For numerical stability, if x > (α+1)/(α+β+2) the symmetry relation I_x(α,β) = 1 − I_{1−x}(β,α) is applied first.
What is the Beta function B(α, β)?
B(α, β) = Γ(α)Γ(β)/Γ(α+β) is the normalizing constant that makes the PDF integrate to 1. For integer values, Γ(n) = (n−1)!, so B(2,3) = 1!×2!/3! = 1/12. For non-integer values, the Gamma function is computed using the Lanczos approximation.
How does skewness relate to α and β?
The skewness of Beta(α,β) is 2(β−α)√(α+β+1) / ((α+β+2)√(αβ)). When α = β the distribution is symmetric (skewness = 0). When α < β the distribution is right-skewed (positive skewness, longer right tail). When α > β it is left-skewed (negative skewness). The skewness magnitude decreases as α and β both increase toward infinity.
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