Calculator Inputs
Example Data Table
| Scenario | α | β | x₁ | x₂ | p | Interpretation |
|---|---|---|---|---|---|---|
| Left-skewed reliability prior | 5.00 | 2.00 | 0.40 | 0.80 | 0.90 | Higher mass sits near stronger outcomes. |
| Right-skewed risk proportion | 2.50 | 6.00 | 0.10 | 0.45 | 0.95 | Most probability remains near lower proportions. |
| Symmetric bounded uncertainty | 3.00 | 3.00 | 0.25 | 0.75 | 0.50 | Distribution centers around the midpoint. |
Formula Used
Density: f(x) = xα-1(1-x)β-1 / B(α,β), for 0 ≤ x ≤ 1.
Beta function: B(α,β) = Γ(α)Γ(β) / Γ(α+β).
Cumulative probability: F(x) = Ix(α,β), the regularized incomplete beta function.
Mean: α / (α + β).
Variance: αβ / [(α+β)2(α+β+1)].
Mode: (α-1)/(α+β-2), when α > 1 and β > 1.
Quantile: solve F(x) = p numerically by bisection.
This page uses gamma-based logarithms, a continued fraction for stable cumulative evaluation, and iterative inversion for target quantiles.
How to Use This Calculator
- Enter positive values for alpha and beta.
- Set x₁ as the point for density and cumulative output.
- Set x₂ to estimate the interval probability from x₁ to x₂.
- Choose probability p when you need a matching quantile.
- Press the calculate button to place results above the form.
- Export the computed summary as CSV or PDF if needed.
Why the Beta Distribution Matters
The beta distribution models uncertain proportions, rates, probabilities, and bounded shares. It is flexible enough to represent symmetric, skewed, flat, or sharply peaked behavior with only two parameters. That makes it valuable in Bayesian analysis, quality estimation, conversion modeling, reliability work, and risk assessment.
When alpha exceeds beta, the distribution leans toward larger values. When beta exceeds alpha, it leans smaller. Equal values produce symmetry. Because the support stays between zero and one, it fits percentages and probabilities better than many unrestricted distributions.
FAQs
Alpha mainly influences how strongly mass moves toward 1. Larger alpha values increase central or right-side concentration, depending on beta.
Beta mainly influences how strongly mass moves toward 0. Larger beta values increase left-side concentration, depending on alpha.
Yes. Convert percentages to decimals first. For example, 35% becomes 0.35 before using x-values or reading quantiles.
If alpha or beta is not greater than 1, the interior mode formula does not apply. The shape can peak at a boundary instead.
It is the probability that X falls between x₁ and x₂. The calculator computes it as F(x₂) minus F(x₁).
The inverse cumulative value is found with repeated bisection. That approach is stable and usually very accurate for practical probability work.
Use it for probabilities, proportions, defect rates, completion shares, or any bounded variable restricted to the interval from 0 to 1.
Skewness shows directional asymmetry. Positive skew indicates a longer right tail, while negative skew indicates a longer left tail.