Gamma Distribution Probability Calculator｜PDF, CDF and Survival from Shape k and Scale θ
Enter the shape parameter k, the scale parameter θ and a point x to get the density f(x), the cumulative probability F(x) and the survival 1−F(x) at once. Mean, variance, standard deviation and mode sit in the same panel, so you can read tail probabilities for waiting times and lifetimes without leaving the page.
💡 About this tool
The gamma distribution models positive-valued, right-skewed continuous quantities: the time until the k-th event in a Poisson process, the lifetime of a component before failure, the total rainfall over a period. Whenever you know the shape k and the scale θ, this calculator turns them into a point density and tail probability.
By hand, the density f(x) = x^(k-1) e^(-x/θ) / (θ^k Γ(k)) carries the gamma function Γ(k), which is awkward when k is non-integer. The tool routes Γ(k) through a log-gamma (Lanczos) approximation and computes the CDF with the regularized lower incomplete gamma, switching between a series expansion and a continued fraction at the boundary. That keeps results stable across small and large k without overflow.
For SREs estimating burst intervals, reliability engineers analysing time-to-failure across multiple stages, or inventory planners sizing replenishment lead time, the survival figure is broken out separately so "probability of being at most x" and "probability of exceeding x" never get confused.
🧐 Frequently Asked Questions
What do shape k and scale θ control? Shape k sets the skew of the curve and scale θ stretches it horizontally. Larger k pushes the shape toward symmetry; larger θ spreads the distribution out and lengthens the typical waiting time. The mean is kθ and the variance is kθ².
I work with a rate, not a scale. How do I enter it? In the rate parametrization with rate β, the scale is θ = 1/β. If you think in rates, enter 1/β as θ. The parametrization in use is also stated under the formula note.
Can the PDF exceed 1? Yes. For a continuous distribution, f(x) is a density, not a probability, so it can rise above 1 when θ is small. To read an actual probability, use the cumulative F(x) or the survival 1−F(x) instead.
What happens at k = 1? A gamma distribution with shape k = 1 reduces to an exponential distribution with mean θ. It then describes the waiting time until the first event.
Why does the mode show 0? The mode is computed as (k−1)θ when k ≥ 1. For k below 1 the density is largest near x = 0, so the tool reports 0.
📚 Gamma, Erlang and the telephone exchange
When the shape k is a positive integer, the gamma distribution is also known as the Erlang distribution: the sum of k independent exponential waiting times, describing how long until the k-th event arrives. Set k = 1 and it collapses back to a single exponential wait.
The Erlang name comes from Agner Krarup Erlang, who studied the number of simultaneous calls hitting a telephone exchange in early-twentieth-century Denmark. That queueing problem seeded an entire field: today the same distribution sizes call-center staffing, models time-to-failure for multi-stage systems, and underpins how engineers reason about waiting in any multi-server queue. It is a neat reminder that a workhorse of modern reliability theory began at a switchboard.