Calculator Inputs
The page stays in a single-column flow, while the form fields adapt to 3 columns on large screens, 2 on small screens, and 1 on mobile.
Example Data Table
This sample demonstrates one practical setup using shape α = 4 and scale θ = 1.5.
| Example Parameter | Value | Computed Output | Result |
|---|---|---|---|
| Shape α | 4.000000 | Mean | 6.000000 |
| Scale θ | 1.500000 | Variance | 9.000000 |
| Point x | 3.500000 | PDF at x | 0.136878 |
| Interval [2, 6] | 2.000000 to 6.000000 | Interval Probability | 0.520036 |
| Percentile | 90.000000% | 90th Percentile Value | 10.021175 |
Formula Used
Probability Density Function
For shape α and scale θ, the density is f(x) = xα-1 e-x/θ / (Γ(α) θα), for x ≥ 0.
Cumulative Distribution Function
The cumulative probability is F(x) = γ(α, x/θ) / Γ(α), where γ(α, x/θ) is the lower incomplete gamma function.
Rate Form
If rate β is used instead of scale, then θ = 1 / β and the density becomes f(x) = βα xα-1 e-βx / Γ(α).
Common Summary Measures
Mean: αθ
Variance: αθ²
Standard deviation: √α · θ
Mode: (α − 1)θ, when α ≥ 1
Skewness: 2 / √α
Excess kurtosis: 6 / α
Interval and Percentile
P(a ≤ X ≤ b): F(b) − F(a)
Quantile xp: solve F(xp) = p
How to Use This Calculator
- Select whether you want to enter the distribution using scale θ or rate β.
- Enter a positive shape α and the matching positive scale or rate value.
- Provide a point x for density and cumulative probability evaluation.
- Enter lower and upper bounds to measure probability across an interval.
- Type a percentile such as 90 to estimate the corresponding quantile value.
- Choose graph resolution and number formatting precision.
- Press the calculate button to display results above the form.
- Use CSV or PDF export to save the calculation summary.
FAQs
1. What does the gamma distribution model?
It models positive continuous values, especially waiting times, lifetimes, and accumulated random processes. It is useful when data are skewed and cannot go below zero.
2. What is the difference between scale and rate?
They are reciprocal parameterizations. Scale is θ, rate is β, and θ = 1/β. Both describe the same distribution when paired with the same shape α.
3. Why is the mode sometimes unavailable?
When shape α is less than 1, the density peaks at zero and does not have an interior mode in the usual positive range. The calculator shows N/A for that case.
4. What does the CDF value mean?
CDF at x means the probability that the random variable is less than or equal to x. It is the total shaded area under the density curve up to that point.
5. What is survival probability?
Survival probability is 1 − CDF(x). It measures the chance that the variable exceeds x, which is helpful for risk, reliability, and waiting-time analysis.
6. How is the percentile value found?
The calculator numerically searches for the x-value whose cumulative probability matches the requested percentile. This inverse lookup is often called the quantile function.
7. Why does the graph sometimes rise sharply near zero?
For shape α below 1, the gamma density becomes very steep near zero. That behavior is mathematically expected and reflects strong concentration near the lower boundary.
8. When should I use the gamma distribution instead of normal?
Use gamma when values are strictly positive and right-skewed. The normal distribution works better for symmetric data that can reasonably extend around a central mean.