Gamma Distribution

 

Gamma Distribution 

The Gamma distribution is a continuous probability distribution used to model:

• ⏱ waiting time until multiple events occur

• 📊 positive-valued quantities (time, rainfall, insurance claims, lifetimes)

• 🧠 Bayesian statistics (conjugate prior for Poisson, Exponential rates)

It generalizes the Exponential distribution.

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Definition

A random variable $X$ follows a Gamma distribution:

$$X \sim \text{Gamma}(\alpha,\beta)$$

Probability density function (rate form)

$$f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, \quad x>0$$

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Parameters

Symbol	Meaning
$\alpha$	shape parameter
$\beta$	rate parameter
$\Gamma(\alpha)$	Gamma function (generalized factorial)

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Gamma function intuition

$$\Gamma(n)=(n-1)! \quad \text{for integer } n$$

Mean and variance

For $X\sim\text{Gamma}(\alpha,\beta)$

$$E[X]=\frac{\alpha}{\beta}$$ $$\text{Var}(X)=\frac{\alpha}{\beta^2}$$

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Intuition: Waiting for events

Imagine:

• Events occur randomly at rate $\beta$
  

• You wait until α events happen

👉 That waiting time follows Gamma(α, β).

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Special case

If $\alpha=1$:

$$\text{Gamma}(1,\beta)=\text{Exponential}(\beta)$$

So exponential = waiting for first event.

Gamma = waiting for α-th event.

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 Example 1 — Machine failures

Suppose failures occur at 2 per hour.

$$\beta=2$$

Time until 3rd failure:

$$X\sim \text{Gamma}(3,2)$$

Mean waiting time:

$$E[X]=\frac{3}{2}=1.5 \text{ hours}$$

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Example 2 — Rainfall accumulation

Let rainfall intensity be random.

If total rain collected over time follows:

$$X\sim\text{Gamma}(5,1)$$

Then:

$$E[X]=5$$
$$\text{Var}(X)=5$$

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Shape behavior

When α < 1

• sharply decreasing

• heavy near zero

When α = 1

• exponential decay

When α > 1

• bell-shaped but skewed right

As α increases:

• distribution becomes more symmetric

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Important properties

✅ Sum of Gamma variables

If

$$X_1\sim \text{Gamma}(\alpha_1,\beta),\quad X_2\sim \text{Gamma}(\alpha_2,\beta)$$

independent, then

$$X_1+X_2\sim \text{Gamma}(\alpha_1+\alpha_2,\beta)$$

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✅ Relation to Chi-square

$$\chi^2_k \sim \text{Gamma}\left(\frac{k}{2},\frac{1}{2}\right)$$

Used in hypothesis testing.

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✅ Bayesian statistics (very important)

Gamma is conjugate prior for:

• Poisson rate λ

• Exponential rate β

If:

$$\lambda\sim\text{Gamma}(\alpha,\beta)$$

and data are Poisson, posterior is also Gamma.

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Summary

Distribution	What it models
Exponential	wait for first event
Gamma	wait for α events
Poisson	count events in time

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Python example

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

x = np.linspace(0,10,400)

plt.plot(x, gamma.pdf(x, a=2, scale=1))
plt.plot(x, gamma.pdf(x, a=5, scale=1))
plt.plot(x, gamma.pdf(x, a=9, scale=1))

plt.title("Gamma distribution for different shape parameters")
plt.show()

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✔️ Final takeaway

Gamma distribution is used when:

• values are continuous and positive

• modeling waiting time

• Bayesian inference for rates

• reliability and queueing systems

It generalizes exponential and connects to Poisson processes.