Gamma Distribution

Definition

Core Statement

The Gamma Distribution models the time until $k$ events occur in a Poisson process. It is a generalization of the Exponential Distribution (which models time until 1 event). It is defined by shape parameter $k$ (or $α$ ) and scale parameter $θ$ (or rate $β$ ).

Purpose

Waiting Times: How long until 5 customers arrive? How long until the 3rd component fails?
Reliability Engineering: Modeling fatigue life where damage accumulates.
Bayesian Statistics: Conjugate prior for the rate parameter of a Poisson or Exponential likelihood.
Financial Modeling: Asset sizes or insurance claims (skewed, positive data).

Relations (The Family Tree)

Exponential: Gamma with shape $k = 1$ . (Wait for 1st event).
Chi-Square: Gamma with specific parameters ( $α = ν / 2, β = 1 / 2$ ).
Erlang: Gamma where $k$ is an integer.

Worked Example: Multiple Failures

Problem

A server crashes on average once every 2 days ( $λ = 0.5$ per day).
Question: What is the probability that the 3rd crash happens within 10 days?

Setup:

Events to wait for ( $k$ or $α$ ): 3.
Rate ( $λ$ or $β$ ): 0.5.
This is Gamma(3, 0.5).

Calculation:
Using Python or Tables ( $P (X \leq 10)$ ):

Result \approx 0.875

Intuition: Since average time per crash is 2 days, 3 crashes take ~6 days. So 10 days is plenty of time. 87.5% chance.

Parameters

Warning: Different fields use different parametrizations!

Shape-Scale ( $k, θ$ ): Mean = $k θ$ . (Engineering).
Shape-Rate ( $α, β$ ): Mean = $α / β$ . (Bayesian/Math).
- $β = 1 / θ$ .

Python Implementation

from scipy.stats import gamma
import matplotlib.pyplot as plt

# Parameters (using 'a' as shape alpha)
alpha = 3  # Wait for 3 events
loc = 0
scale = 2  # Mean time per event (theta = 1/beta) (If rate=0.5, scale=2)

rv = gamma(alpha, loc=loc, scale=scale)

# Probability wait < 10
prob = rv.cdf(10)
print(f"Prob < 10: {prob:.4f}")

# Plot
import numpy as np
x = np.linspace(0, 20, 100)
plt.plot(x, rv.pdf(x))
plt.title("Gamma Distribution (k=3, theta=2)")
plt.show()

Exponential Distribution - Special case ( $k = 1$ ).
Poisson Distribution - Counts events in fixed time (Dual concept).
Chi-Square Distribution - Another special case.
Beta Distribution - Associated conjugate prior.