ガンマ分布

ガンマ分布

ガンマ分布は、二パラメータ族の連続 確率分布. It is defined by a shape parameter (k) and a scale parameter (θ). The probability ガンマ分布の確率密度関数（PDF）は次のように表されます：

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.

この分布は、さまざまな分野で特に役立ちます statistics, queuing models, and 機械学習, 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 指数分布.

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.

機械学習では、ガンマ分布は ベイズ推論 and for modeling uncertainty. Its ability to express a wide range of shapes makes it a flexible choice for various データモデリングに利用されます シナリオ