What Is Gamma Distribution?
The gamma distribution is a flexible model for positive, right-skewed quantities such as waiting times and rainfall amounts. It has a shape parameter k and a scale parameter θ, and the equivalent rate β = 1/θ is shown too, since textbooks and software use both conventions.
The calculator returns the cumulative probability, the probability between two values, the density, the median, the mode, the 5th and 95th percentiles, the mean kθ, the variance kθ² and the skewness and excess kurtosis. The curve shades the probability you asked for.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as scale versus rate and the link with the exponential and chi-square distributions.
How to Use the Gamma Distribution Calculator
- Enter the shape k.
- Choose whether you are supplying the scale θ or the rate β, and enter it. Check which your textbook or software uses.
- Enter x, and optionally an upper bound for an interval probability.
- Read the probabilities and the percentile table.
Gamma Distribution Formula
Density
Cumulative distribution
Moments
Gamma Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| k | Shape parameter | k > 0 |
| θ | Scale parameter | θ > 0 |
| β | Rate parameter | 1 ÷ θ |
| μ | Mean | kθ |
| σ² | Variance | kθ² |
Gamma Distribution Worked Example
Using the example values (press “Load example” above to try them):
- Shape k = 3, scale θ = 2 (rate β = 0.5).
- Density: f(x) = x^(k−1) e^(−x/θ) ÷ (Γ(k) θ^k); at x = 5 this is 0.12825781.
- Cumulative probability P(X ≤ x) = P(k, x/θ) = 0.45618688, where P is the regularised lower incomplete gamma function.
- Mean = kθ = 6; variance = kθ² = 12.
Result: P(X ≤ 5) = 0.45618688; P(X > 5) = 0.54381312.
Shape, scale and rate: avoiding parameter mix-ups
The gamma has two common parameterisations. In the shape-scale form (used by many textbooks and by R’s scale argument) the mean is kθ. In the shape-rate form (used by R’s default, by Excel’s GAMMA.DIST with beta = scale, and in Bayesian work) the mean is k/β. Entering the wrong one changes every result, so check what your source calls its second parameter.
| Source | Parameters | Mean |
|---|---|---|
| Most textbooks | shape k, scale θ | kθ |
| Bayesian statistics | shape α, rate β | α/β |
| R dgamma | shape, rate (or scale=) | shape/rate |
| Excel GAMMA.DIST | alpha, beta (scale) | alpha × beta |
| SciPy gamma | a, scale | a × scale |
Relationships with other distributions
If waiting times between events are exponential, the time to the k-th event is gamma with shape k. This links the gamma to the Poisson process and, through the chi-square, to statistical tests.
| Distribution | Gamma form |
|---|---|
| Exponential with mean θ | k = 1 |
| Chi-square with ν df | k = ν/2, θ = 2 |
| Erlang with k stages | Integer k |
| Normal for large k | Approximately normal as k grows |
Typical applications
- Reliability and queueing: time until the k-th failure or arrival.
- Insurance: sizes of claims.
- Hydrology: rainfall totals.
- Bayesian statistics: conjugate prior for a Poisson rate.
- Medicine: lengths of stay and survival times.
Where Gamma Distribution Is Used
Waiting-time models
Modelling the time until several events have occurred.
Positive skewed data
Modelling amounts that cannot be negative and have a long right tail.
Percentiles
Finding cut-offs such as the 95th percentile.
Coursework
Checking probabilities and moments for given k and θ.
Common Gamma Distribution Mistakes to Avoid
- Check which parameterisation your formulas use.
- The mode formula applies only for k ≥ 1.
- For large k the distribution looks like a normal curve.
Gamma Distribution Terms and Definitions
- Shape
- The parameter controlling skew and form of the curve.
- Rate
- The reciprocal of the scale; the events-per-unit-time view.
- Incomplete gamma function
- The special function whose normalised form gives the gamma CDF.
Gamma Distribution FAQs
What is the gamma distribution used for?
Modelling positive, right-skewed quantities such as waiting times until the k-th event, claim sizes and rainfall.
What is the difference between the scale and rate parameters?
They are reciprocals: rate β = 1/θ. The mean is kθ with scale and k/β with rate.
How is the gamma related to the exponential?
An exponential distribution is a gamma with shape k = 1. A gamma with integer shape k is the sum of k independent exponentials.
How is the gamma related to chi-square?
A chi-square distribution with ν degrees of freedom is a gamma with shape ν/2 and scale 2.
How do I find a percentile of a gamma distribution?
Use the inverse CDF. The percentile table above lists common ones for your parameters.
Can the shape be less than 1?
Yes. The density then goes to infinity at 0 and decreases steadily.