Model uncertainty with flexible shapes and precise probabilities. Compare tails, intervals, and percentiles in seconds. See curves, export findings, and explain results with confidence.
Enter positive shape parameters, a target x value, an interval range, and a percentile probability. The form uses three columns on large screens, two on medium screens, and one on mobile.
These examples show how different shape parameters change density and cumulative probability across the unit interval.
| Case | Alpha | Beta | x | CDF | Interpretation | |
|---|---|---|---|---|---|---|
| Uniform | 1.0 | 1.0 | 0.50 | 1.0000 | 0.5000 | Equal weight across the interval. |
| Symmetric peak | 2.0 | 2.0 | 0.50 | 1.5000 | 0.5000 | Mass concentrates near the center. |
| Right-skewed | 2.0 | 5.0 | 0.40 | 1.5552 | 0.7667 | More probability sits near smaller values. |
| Left-skewed | 5.0 | 2.0 | 0.70 | 2.1609 | 0.4202 | More probability sits near larger values. |
f(x) = xα-1(1-x)β-1 / B(α,β), for 0 ≤ x ≤ 1 and α, β > 0
B(α,β) = Γ(α)Γ(β) / Γ(α+β)
F(x) = Ix(α,β), where Ix is the regularized incomplete beta function.
P(L ≤ X ≤ U) = F(U) − F(L)
Mean = α / (α + β)
Variance = αβ / [(α + β)2(α + β + 1)]
Mode = (α − 1) / (α + β − 2), when α > 1 and β > 1
It models uncertainty for values limited to the interval from 0 to 1. Common examples include probabilities, proportions, rates, completion ratios, and Bayesian posterior beliefs.
The beta distribution is defined only on the unit interval. Any valid probability or proportion must remain within those natural lower and upper limits.
They control shape, concentration, skewness, and boundary behavior. Larger values often create tighter concentration, while unequal values shift the mass toward one side.
The PDF gives relative density at a single point. The CDF gives the accumulated probability up to that point, which is more useful for tail and interval questions.
It measures how much total probability lies between your chosen lower and upper bounds. This is useful when evaluating acceptable ranges or confidence regions.
The quantile is the x value whose cumulative probability equals your chosen percentile p. For example, p = 0.95 returns the 95th percentile location.
When alpha or beta falls below 1, the curve can spike near a boundary. That behavior is valid and indicates heavy concentration close to 0 or 1.
It is useful in Bayesian analysis, A/B testing, reliability studies, quality control, finance, risk modeling, and any situation involving bounded proportions.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.