Gamma Distribution Calculator

Enter the shape parameter (α) and scale parameter (β) along with an x value to compute Gamma Distribution probabilities. Your results include the PDF (probability density), CDF (cumulative probability), P(X < x), and P(X > x) — plus a visual density curve to see where your value falls on the distribution.

The shape parameter α must be greater than 0.

The scale parameter β must be greater than 0. Use rate = 1/β if working with rate parameterization.

Choose whether β is entered as scale or rate (λ). If rate, β is set to 1/λ internally.

The value at which to evaluate the distribution. Must be ≥ 0.

Results

P(X < x) — Cumulative Probability

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PDF — Probability Density f(x)

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P(X > x) — Upper Tail Probability

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Mean (α·β)

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Variance (α·β²)

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Standard Deviation

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Mode ((α−1)·β for α≥1)

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Gamma Distribution Density Curve

Results Table

Frequently Asked Questions

What is the Gamma distribution used for?

The Gamma distribution is a continuous probability distribution commonly used to model waiting times, reliability analysis, and insurance claim sizes. It generalizes the exponential distribution and is widely applied in Bayesian statistics, queuing theory, and hydrology.

What is the difference between the shape (α) and scale (β) parameters?

The shape parameter α controls the skewness and overall form of the distribution — higher α values make the distribution more symmetric and bell-shaped. The scale parameter β stretches or compresses the distribution along the x-axis, effectively setting the scale of the random variable.

What is the difference between scale and rate parameterization?

Some textbooks define the Gamma distribution using a rate parameter λ = 1/β instead of the scale β. Both forms describe the same family of distributions. If you have a rate λ, simply enter it and select 'Rate' — the calculator converts it to scale internally using β = 1/λ.

What does PDF mean in the context of the Gamma distribution?

The PDF (Probability Density Function) gives the relative likelihood of the random variable taking a specific value x. For the Gamma distribution, f(x) = x^(α−1) · e^(−x/β) / (β^α · Γ(α)). Note that the PDF is not a probability itself, but its integral over an interval gives probability.

What does CDF (cumulative distribution function) represent?

The CDF, P(X < x), gives the probability that the random variable X takes a value less than or equal to x. It is computed as the integral of the PDF from 0 to x and always returns a value between 0 and 1.

How do I compute P(X > x) using this calculator?

P(X > x) is the upper tail probability and equals 1 − P(X < x). The calculator outputs this directly in the 'Upper Tail Probability' result field so you don't need to subtract manually.

What are the mean and variance of the Gamma distribution?

For a Gamma(α, β) distribution using scale parameterization, the mean is α·β and the variance is α·β². The standard deviation is √(α)·β. These summary statistics are shown directly in the calculator's output.

When does the Gamma distribution reduce to an Exponential or Chi-squared distribution?

When α = 1, the Gamma distribution reduces to an Exponential distribution with mean β. When β = 2 and α = k/2 for integer k, it becomes a Chi-squared distribution with k degrees of freedom. The calculator works for any positive α and β, covering all these special cases.

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