What is the shape parameter k?

The shape parameter k (also written as alpha) determines the form of the distribution. When k = 1 the Gamma reduces to the exponential distribution. When k is an integer it reduces to the Erlang distribution. Larger k values produce a more symmetric, bell-shaped curve.

What is the scale parameter theta?

The scale parameter θ (theta) stretches the distribution along the x-axis without changing its fundamental shape. Its reciprocal β = 1/θ is called the rate parameter and is used in some textbook formulations.

How is the CDF calculated?

The CDF is the regularized lower incomplete gamma function P(k, x/θ) = γ(k, x/θ) / Γ(k), where γ is the lower incomplete gamma function. This calculator uses a series expansion for accurate numerical evaluation.

What is the mode of the Gamma distribution?

The mode is (k - 1) · θ for k ≥ 1, and 0 for 0 < k < 1. When k = 1 the mode is 0, consistent with the exponential distribution. When k < 1 the PDF is strictly decreasing from infinity.

How does the Gamma relate to the chi-squared distribution?

A chi-squared distribution with n degrees of freedom is a special case of the Gamma distribution with k = n/2 and θ = 2.

What is the skewness of the Gamma distribution?

The skewness is 2 / √k. It is always positive (right-skewed). As k increases, the skewness decreases toward 0 and the distribution approaches a normal shape by the central limit theorem.

Gamma Distribution Calculator

Enter shape (k) and scale (θ) parameters plus a value x to compute the Gamma distribution PDF, CDF, and key statistics instantly.

Parameters

Shape Parameter k (must be > 0)

Controls the shape; k = 1 gives the exponential distribution.

Scale Parameter θ (must be > 0)

Stretches the distribution; rate β = 1/θ.

Value x (≥ 0)

The point at which to evaluate the distribution.

Quick Presets

Results at x

Enter parameters and press Calculate.

Distribution Statistics

Enter parameters and press Calculate.

Formulas

PDF:: f(x) = x^k−1 e^−x/θ / (θ^k Γ(k))
CDF:: F(x) = P(k, x/θ) — regularized incomplete gamma
Mean:: k · θ
Variance:: k · θ²
Mode:: (k − 1) · θ for k ≥ 1, else 0
Skewness:: 2 / √k

Summary

The Gamma distribution is a two-parameter family of continuous probability distributions that generalizes the exponential and chi-squared distributions. Given shape parameter k (> 0) and scale parameter θ (> 0), this calculator computes the probability density function (PDF), cumulative distribution function (CDF), and key statistics including mean, variance, mode, and standard deviation. Results update on demand as you enter values.

How it works

Enter a positive shape parameter k — controls the skewness and peak shape of the distribution.
Enter a positive scale parameter θ (theta) — stretches or compresses the distribution along the x-axis.
Enter a non-negative value x at which to evaluate the distribution.
The calculator computes the PDF: x^(k-1) · e^(-x/θ) / (θ^k · Γ(k)).
The CDF is computed via the regularized incomplete gamma function P(k, x/θ).
Distribution statistics (mean, variance, mode, skewness) are shown in the summary panel.

Use cases

Modeling waiting times for the k-th event in a Poisson process.
Insurance and finance: modeling aggregate claim sizes and loss distributions.
Bayesian statistics: using the Gamma as a conjugate prior for the Poisson rate.
Hydrology: fitting rainfall amounts and flood discharge data.
Reliability engineering: modeling component lifetimes with non-constant hazard rates.
Machine learning: prior distributions in probabilistic graphical models.
Biology: modeling cell cycle times and neural inter-spike intervals.
Statistics coursework: verifying manual calculations for homework or exams.

Frequently Asked Questions

Last updated: 2026-05-25 · Reviewed by Nham Vu