Gamma Distribution Alpha Beta Calculator
Example Data Table
| Alpha | Beta | Beta Meaning | x | CDF | Mean | Variance | |
|---|---|---|---|---|---|---|---|
| 2 | 1.5 | Rate | 2 | 0.224042 | 0.800852 | 1.333333 | 0.888889 |
| 3.5 | 0.8 | Rate | 4 | 0.179741 | 0.506105 | 4.375 | 5.46875 |
| 4 | 2 | Scale | 6 | 0.112021 | 0.352768 | 8 | 16 |
| 0.9 | 1.2 | Rate | 1 | 0.332111 | 0.737231 | 0.75 | 0.625 |
Formula Used
This page supports both common gamma parameterizations. Alpha is the shape parameter. Beta can represent rate or scale.
When beta means rate
f(x) = [β^α / Γ(α)] x^(α-1) e^(-βx)
F(x) = P(α, βx)
Mean = α / β
Variance = α / β²
Mode = (α - 1) / β for α ≥ 1
When beta means scale
f(x) = x^(α-1) e^(-x/β) / [Γ(α) β^α]
F(x) = P(α, x / β)
Mean = αβ
Variance = αβ²
Mode = (α - 1)β for α ≥ 1
Here Γ(α) is the gamma function,
and P(α, z) is the regularized lower incomplete gamma function.
How to Use This Calculator
- Enter alpha as the shape parameter.
- Enter beta and choose whether it means rate or scale.
- Enter the x value for density and cumulative probability.
- Optionally set a custom graph maximum and point count.
- Press calculate to view the result above the form.
- Use the export buttons to save CSV or PDF reports.
Frequently Asked Questions
1) What do alpha and beta represent?
Alpha controls shape. Beta can represent rate or scale, depending on the selected option. Rate makes larger beta concentrate values nearer zero. Scale stretches the distribution rightward as beta grows.
2) When should I choose rate or scale?
Use rate when formulas use e-βx. Use scale when formulas use e-x/β. They describe the same family after converting rate = 1 / scale.
3) Can x be negative?
No. The gamma distribution is defined for x ≥ 0. Negative inputs are invalid, so the calculator blocks them.
4) Why does mode show boundary at 0?
When alpha is below 1, the density decreases from the left boundary. The highest density occurs at x = 0, so there is no positive interior mode.
5) What does the CDF mean?
The CDF gives Pr(X ≤ x). It measures the accumulated probability from zero up to your selected x.
6) Why might the graph spike near zero?
For alpha below 1, gamma density can rise sharply near zero. The chart keeps the curve readable by sampling positive values along the range.
7) Can I use this for waiting times or reliability?
Yes. Gamma models waiting times, lifetimes, rainfall totals, claim sizes, and other positive skewed variables when a flexible right-tailed model is useful.
8) Are exports exact?
Exports include the displayed summary values and plotted series generated on the page. Small rounding differences can appear because tables show formatted numbers.