Gamma Distribution

Probability · Axiom Academy

A flexible family generalizing the exponential distribution Before defining the Gamma distribution, we need the Gamma function Γ(α), which generalizes the factorial to non-integer values. For positive integers n, we have Γ(n) = (n-1)!. This function appears in the normalization constant of the distribution. Key property: Γ(α + 1) = α · Γ(α), which mirrors the factorial property n! = n · (n-1)! 2. Probability Density Function The Gamma PDF has two parameters: α (shape) controls the shape of the distribution, and λ (rate) controls the scale. Different values of α produce different shapes - from exponential decay (α = 1) to bell-shaped curves (α > 1). A key insight: If X₁, X₂, ..., Xₙ are independent Exp(λ) random variables, then their sum follows a Gamma(n, λ) distribution. This makes the Gamma distribution perfect for modeling the total waiting time for n events in a Poisson process.

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