How to read the table: each cell shows P(X ≤ x) for the selected parameters α and β. The row gives the first decimal digit of x; the column adds the second (×0.01). The range is x ∈ [0.00, 0.99].
Click a cell to read the value.
P(X ≤ x) for x from 0.00 to 0.99
How to read the table: each cell shows f(x) = x^(α−1) · (1−x)^(β−1) / B(α,β). For α < 1 or β < 1 the density may diverge at the endpoints (very large values are shown as "∞").
Click a cell to read the value.
f(x) for x from 0.00 to 0.99
How to read the table: each cell shows the quantile x such that P(X ≤ x) = p. Rows: different values of α. Columns: cumulative probabilities p. Select β with the selector.
Click a cell to read the value.
Quantiles Be⁻¹(p; α, β) — P(X ≤ x) = p
How to use these tables
Beta Distribution
The Beta distribution Be(α, β) has support on [0, 1] and is very flexible for modeling proportions and probabilities. Its density function is:
\( f(x;\alpha,\beta) = \dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad x \in (0,1) \)
where B(α, β) = Γ(α)Γ(β)/Γ(α+β) is the beta function. The mean is μ = α/(α+β) and the variance σ² = αβ / [(α+β)²(α+β+1)].
Shapes of the distribution
- α = β = 1: Uniform(0,1) distribution.
- α = β > 1: unimodal and symmetric around 0.5.
- α > β: left-skewed (mean > 0.5).
- α < β: right-skewed (mean < 0.5).
- α < 1 or β < 1: U-shaped (bimodal at the endpoints).
Relationship with F and t
If F ~ F(2α, 2β) then X = αF/(β + αF) ~ Be(α, β). The regularized incomplete beta function is also the CDF of the F distribution and of Student's t distribution.