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

Name: Gamma Distribution Calculator
Availability: InStock
Author: Nham Vu

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

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

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

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Last updated: 2026-05-23 · Reviewed by Nham Vu