---
title: "Gamma distribution: characteristics, applications"
subject: "Probability and Statistics"
chapter: 3
key: "semester_5/probability_and_statistics:gamma-definition-characteristics"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-definition-characteristics
site: BCT Engineering
---

# Gamma distribution: characteristics, applications

Probability and Statistics · Chapter 3: Continuous Probability Distributions

- Asked in: 2075 Chaitra, 5 marks
- Video: [The gamma distribution](https://www.youtube.com/watch?v=cpW40zPdAQ8) — Equitable Equations
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/gamma-definition-characteristics
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-3

## Question

Define Gamma distribution, chief characteristics and write its applications.

## Short answer

Gamma distribution: pdf f(x)=x^(α-1)e^(-x/β)/(β^α Γ(α)), x>0, mean αβ, variance αβ². It is positively skewed; special cases exponential (α=1) and chi-square (α=ν/2, β=2). Used for waiting times, lifetimes, hydrology, insurance.

## Answer

#### Definition
A continuous random variable X has a **gamma distribution** with shape parameter α > 0 and scale parameter β > 0 if its pdf is

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

where Γ(α) = ∫₀^∞ t^(α−1) e^(−t) dt is the gamma function (Γ(n) = (n − 1)! for integer n, Γ(1/2) = √π).

#### Chief characteristics
1. Defined for x > 0 only; it is positively skewed.
2. Mean = αβ, variance = αβ², so standard deviation = β√α.
3. Mode = (α − 1)β for α ≥ 1.
4. The shape depends on α: for α = 1 it reduces to the **exponential** distribution with mean β; for α = ν/2 and β = 2 it is the **chi-square** distribution with ν degrees of freedom.
5. As α grows large, it tends to a normal shape.
6. When α is a positive integer it models the waiting time until the α-th event of a Poisson process (Erlang distribution).
7. The sum of independent gamma variables with the same β is again gamma, with the α's added.
8. The mgf is (1 − βt)^(−α), t < 1/β.

#### Applications
- Waiting time or time to failure until the α-th event (queues, reliability of systems with standby units).
- Service times and lifetimes of components.
- Daily rainfall, river flow and flood levels (hydrology).
- Daily electricity or water consumption.
- Insurance claim sizes and income distribution.
- Through the chi-square case, it underlies tests of variance and goodness of fit.

## Check yourself

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

1. The gamma has mean and variance:
   A. alpha beta and alpha beta^2
   B. alpha/beta and alpha/beta^2
   C. alpha and alpha
   D. beta and beta^2
   Answer: A. Shape alpha, scale beta. (Key point: Parameters)

2. Gamma reduces to exponential when:
   A. alpha = 2
   B. beta = 1 only
   C. alpha = 1
   D. alpha = nu/2
   Answer: C. With alpha = 1, f = e^(-x/beta)/beta. (Key point: Exponential case)

3. Gamma is chi-square with nu df when:
   A. alpha = 2 and beta = nu/2
   B. alpha = nu and beta = 1
   C. alpha = 1 and beta = nu
   D. alpha = nu/2 and beta = 2
   Answer: D. Gives mean nu and variance 2 nu. (Key point: Chi-square case)

4. The gamma is:
   A. Symmetric on all reals
   B. Positively skewed, defined for x > 0
   C. Negatively skewed
   D. Defined for x < 0
   Answer: B. It is a right-skewed waiting-time model. (Key point: Shape)

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