Gamma Distribution¶

Formula¶

\[ f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x},\quad x\ge 0 \]

Plot¶

fn: x*exp(-x)
xmin: 0
xmax: 10
ymin: 0
ymax: 0.5
height: 280
title: Gamma PDF (alpha=2, rate=1)

Parameters¶

\(\alpha>0\): shape
\(\beta>0\): rate (one common parameterization)
\(\Gamma(\alpha)\): gamma function

What it means¶

A flexible positive-valued distribution that generalizes the exponential distribution.

What it's used for¶

Waiting times for multiple Poisson events.
Bayesian priors for rates/precisions.

Key properties¶

Mean \(\alpha/\beta\), variance \(\alpha/\beta^2\) (rate form).
Exponential is the special case \(\alpha=1\).

Common gotchas¶

Scale-vs-rate parameterization varies across sources.
Shape near \(<1\) changes density behavior near zero.

Example¶

The waiting time until the 3rd Poisson event follows a Gamma distribution.

How to Compute (Pseudocode)¶

Input: distribution parameters and query values (for PMF/PDF/CDF or moments)
Output: distribution quantities

validate parameters
for each query value x (or count k):
  evaluate the PMF/PDF/CDF formula from the card
optionally compute moments/statistics from known closed forms or by summation/integration
return the requested values

Complexity¶

Time: Typically \(O(q)\) for \(q\) query values once parameters are known (assuming constant-time formula evaluation per query)
Space: \(O(q)\) for output values (or \(O(1)\) for a single query)
Assumptions: Parameter estimation/fitting cost is excluded; numerical special-function evaluation can affect constants for some families