---
title: "Gamma and chi-squared distributions relation"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:gamma-and-chi-squared-relation"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-and-chi-squared-relation
site: BCT Engineering
---

# Gamma and chi-squared distributions relation

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2074 Chaitra, 2+3 marks
- Video: [Gamma Distributions  - Relationship with the  Chi Square Distribution (Worked Example)](https://www.youtube.com/watch?v=ApP4tecHPTY) — StatsResource
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-and-chi-squared-relation
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

What are Gamma and Chi-squared distributions? Specify relationship between them.

## Short answer

Gamma has pdf x^(α-1)e^(-x/β)/(β^α Γ(α)), mean αβ, variance αβ². Chi-squared with ν df is sum of squares of ν standard normals. Chi-squared is a special case of gamma with α=ν/2, β=2, giving mean ν, variance 2ν.

## Answer

#### Gamma distribution
A continuous variable X > 0 has a gamma distribution with shape α > 0 and scale β > 0 if

f(x) = x^(α−1) e^(−x/β) / (βᵅ Γ(α)), x > 0.

Mean = αβ, variance = αβ². It models waiting times and lifetimes.

#### Chi-squared distribution
A continuous variable χ² > 0 has a chi-squared distribution with ν degrees of freedom if

f(x) = x^(ν/2 − 1) e^(−x/2) / (2^(ν/2) Γ(ν/2)), x > 0.

Mean = ν, variance = 2ν. It arises as the sum of squares of ν independent standard normal variables: χ² = Z₁² + Z₂² + … + Zν². It is used in tests of variance, goodness of fit and independence.

#### Relationship
The chi-squared distribution is a **special case of the gamma distribution** obtained by putting

**α = ν/2 and β = 2**

Substituting these in the gamma pdf gives exactly the chi-squared pdf. Checking the moments: mean αβ = (ν/2)(2) = ν, and variance αβ² = (ν/2)(4) = 2ν.

Also, the exponential distribution is gamma with α = 1, so chi-squared with 2 degrees of freedom is exponential with mean 2.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Chi-squared with nu df is gamma with:
   A. alpha = nu and beta = 2
   B. alpha = nu/2 and beta = 2
   C. alpha = 2 and beta = nu
   D. alpha = 1, beta = nu
   Answer: B. Substituting yields the chi-squared pdf. (Key point: Relationship)

2. A chi-squared variable arises as:
   A. Sum of nu normals
   B. Product of nu normals
   C. Mean of nu normals
   D. Sum of squares of nu independent standard normals
   Answer: D. chi^2 = Z1^2 + ... + Znu^2. (Key point: Chi-square origin)

3. Chi-squared mean and variance are:
   A. nu and 2 nu
   B. nu and nu
   C. 2 nu and nu
   D. nu/2 and nu
   Answer: A. alpha beta = nu, alpha beta^2 = 2 nu. (Key point: Moments)

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