Gammaverteilung

The Gammaverteilung, or gamma distribution, is a continuous probability distribution commonly used in statistics and engineering. It models the waiting times until the occurrence of a specified number of events in a process where events occur randomly and independently at a constant average rate.

The gamma distribution is characterized by two parameters: shape (k or α) and scale (θ or β). The probability density function (PDF) for a gamma-distributed random variable X is given by:

f(x; α, β) = (x^(α - 1) e^(-x/β)) / (β^α Γ(α)) for x > 0,

where Γ(α) represents the gamma function evaluated at α.

The shape parameter determines the form of the distribution, ranging from exponential-like for α = 1 to more symmetric as α increases. The scale parameter stretches or compresses the distribution along the x-axis. When the shape parameter is an integer, the gamma distribution simplifies to the Erlang distribution, often used to model the sum of exponential variables.

The gamma distribution finds applications in various fields, including queuing theory, reliability analysis, and Bayesian statistics. It is particularly useful for modeling waiting times, lifetimes of systems, and the sum of multiple exponential processes.

The gamma distribution's properties include a flexible skewness and the ability to model a wide variety of data shapes, making it a versatile tool in statistical analysis. It is closely related to other distributions, such as the chi-squared and Erlang distributions, which are special cases or functions of the gamma distribution.