Math Problem Statement

Solution

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Math Problem Analysis

Mathematical Concepts

Probability Distribution
Gamma Distribution
Maximum Likelihood Estimation

Formulas

Probability density function of Gamma distribution: f(x; α, λ) = (λ^α x^(α - 1) e^(-λx)) / Γ(α)
Log-likelihood function: log L(α, λ) = nα log λ - n log Γ(α) + (α - 1) Σ log X_i - λ Σ X_i
MLE for λ: λ̂ = nα / Σ X_i

Theorems

Maximum Likelihood Estimation (MLE)
Digamma Function (ψ)

Suitable Grade Level

Graduate Level