How to read the table: each cell shows P(X ≤ x) for the selected shape parameter α and scale parameter β. The row gives the integer part of x; the column adds the first decimal (×0.1).
Click a cell to read the value.
P(X ≤ x) for x from 0.0 to 30.9
How to read the table: each cell shows f(x) = x^(α−1) · e^(−x/β) / (β^α · Γ(α)) for the selected parameters. The density is 0 for x ≤ 0.
Click a cell to read the value.
f(x) for x from 0.0 to 30.9
How to read the table: each cell shows the quantile x such that P(X ≤ x) = p. Rows: different values of α (shape). Columns: cumulative probabilities p. Select β (scale) with the selector.
Click a cell to read the value.
Quantiles Γ⁻¹(p; α, β) — P(X ≤ x) = p
How to use these tables
Gamma Distribution
The Gamma distribution Γ(α, β) generalizes the Exponential distribution. Its density function is:
\( f(x;\alpha,\beta) = \dfrac{x^{\alpha-1}\,e^{-x/\beta}}{\beta^\alpha\,\Gamma(\alpha)}, \quad x > 0, \;\alpha > 0,\; \beta > 0 \)
The mean is μ = αβ and the variance σ² = αβ². The parameter α controls the shape and β the scale.
Special cases
- Γ(1, β): coincides with the Exponential distribution with mean β.
- Γ(ν/2, 2): coincides with the Chi-square distribution with ν degrees of freedom.
- Γ(k, 1) with integer k: sum of k independent Exponentials with mean 1 (Erlang distribution).
Parameterization
This table uses the (α, β) parameterization where β is the scale and 1/β is the rate. Check the parameterization used by your software before comparing values.