Gamma Distribution
The Gamma Distribution is a two-parameter family of continuous probability distributions. It is defined by a shape parameter (k) and a scale parameter (θ). The probability density function (PDF) of the Gamma Distribution is expressed as:
f(x; k, θ) = (1 / (θ^k * Γ(k))) * x^(k-1) * e^(-x/θ), for x > 0, k > 0, and θ > 0, where Γ(k) is the gamma function evaluated at k.
This distribution is particularly useful in various fields such as statistics, queuing models, and machine learning, where it often models waiting times or lifetimes of objects. The Gamma Distribution encompasses several well-known distributions: when k is a positive integer, it becomes the Erlang distribution, and when k = 1, it simplifies to the exponential distribution.
In practical applications, the shape parameter influences the skewness of the distribution, while the scale parameter stretches or compresses the distribution along the x-axis. The mean of the Gamma Distribution is given by μ = k * θ, and the variance is σ² = k * θ².
In machine learning, the Gamma Distribution can be utilized in Bayesian inference and for modeling uncertainty. Its ability to express a wide range of shapes makes it a flexible choice for various data modeling scenarios.