Gamma Distribution — Sum of Exponential Variables
Foundations of Statistics
Flexible Modeling of Positive Data
The gamma distribution extends the exponential to model waiting times for multiple events and right-skewed positive data. Its flexibility makes it ideal for insurance claims, rainfall amounts, and survival analysis.
- Insurance — Modeling claim sizes and aggregate losses in actuarial science
- Meteorology — Predicting rainfall amounts and drought durations
- Healthcare — Survival times in clinical trials and time-to-event data
When data is positive and skewed, the gamma distribution provides the natural framework.
Core Concepts
The gamma distribution generalizes the exponential distribution. It models the waiting time until the -th event in a Poisson process and serves as a flexible model for right-skewed, positive-valued data.
The Gamma Function
Derivation of Mean and Variance
MGF and Additivity
Special Cases
Chi-Squared Connection
Log-Gamma Distribution
Worked Example
Specific Applications
- Insurance and finance — Aggregate claim amounts, ruin probability, and loss modeling.
- Hydrology — Rainfall amounts, flood frequency analysis (often as gamma or Pearson Type III).
- Bayesian statistics — Conjugate prior for the Poisson rate parameter.
- Queueing theory — Erlang distributions model total service time for multiple sequential tasks.