Gamma Distribution : Meaning, Special Cases and Examples

The Gamma distribution is defined for non-negative real numbers and is used to describe the waiting time until a specific event occurs in a Poisson process, the time between events in a Poisson process, and various other continuous, positive, right-skewed phenomena. To formally define the Gamma Distribution, it is necessary to first introduce the gamma function, which is a special mathematical function that provides the normalisation constants used in the probability density function. The gamma function allows the distribution to integrate into one over its positive support, making it a valid probability distribution for modeling positive random variables. An understanding of the gamma function, therefore, lays the groundwork for subsequently exploring the properties and applications of the gamma distribution.

The gamma function depicted by Γ(α), is an extension of the factorial function. The values of the gamma function for non-integer arguments generally cannot be expressed in simple, closed forms. The gamma distribution is written as Gamma (α,λ)

Probability Density Function (PDF) of Gamma Distribution

The PDF of the Gamma distribution is given by:

Mean and Variance of Gamma Distribution

The mean of a random variable is a measure of its central tendency while variance measures the spread or dispersion of a random variable. It tells us how much the values of a random variable deviate from its mean.

1. The mean of the Gamma distribution is 

2. The variance of Gamma Distribution is 

Special Case 1: Exponential Distribution

Gamma with 

The PDF of the exponential distribution is:

Mean and Variance

• The mean of the exponential distribution is 
• The variance is 

Examples of Exponential Distribution

Example 1:

Consider a scenario where the waiting time for a customer to complete a transaction at a bank’s ATM follows an exponential distribution with a scale parameter (λ) of 5 minutes. We want to find the probability that a customer will take less than 3 minutes to complete the transaction.

Solution:

The formula to determine the probability of a customer taking less than 3 minutes to complete the transaction will be: 

In this case, λ = 5, and we want to find P(X < 3), where X is the random variable representing the waiting time.

P(X < 3) = 

P(X < 3) = 

P(X < 3) = 0.99

Therefore, the probability that the customer will take less than 3 minutes to complete the transaction is 0.99.

Example 2:

Suppose the claims at a life insurance company’s 24-hour call centre occur at a 4-per-hour rate. What is the probability that the next call arrives after more than 2 hours?

Solution:

The number of claims (X), in an hour can be expressed as a Poisson distribution with mean as . We need to determine the probability of the next call arriving after more than 2 hours. For this, we will use the following formula:

P(X > 2) = 

P(X > 2) = 

P(X > 2) = 0.0003

Therefore, the probability that the next call will arrive after 2 hours is 0.0003.

Special Case 2: Chi-Square Distribution with Parameter “Degrees of Freedom”

Gamma with , where is a positive integer, and 

The PDF of the chi-square distribution with v degrees of freedom is:

Mean and Variance

• The mean of the chi-square distribution is 
• The variance of chi-square distribution is σ² = 2

Note: A variable with is same as an exponential variable with mean 2.

Examples of Chi-Square Distribution

Example 1:

Suppose we have a sample of 20 individuals, and we want to examine the distribution of their ages. If the ages follow a chi-square distribution with 15 degrees of freedom, find the probability that the total age of the sample is less than 30 years.

Solution:

The PDF of the chi-square distribution is given by, 

In this question, we have to determine the probability that the total age of the taken sample is less than 30 years. For this, we will use the Gamma Distribution table.

Degree of freedom; i.e., = 15, and we have to determine P(X < 30).

P(X < 30) = 0.9881

Example 2:

Consider a study where the number of defects in a batch of products follows a chi-square distribution with 8 degrees of freedom. Find the probability that the number of defects is less than 12.

Solution:

The PDF of the chi-square distribution is given by, 

In this question, we have to determine the probability that the number of defects is less than 30 years. For this, we will use the Gamma Distribution table.

Degree of freedom; i.e., = 8, and we have to determine P(X < 12).

P(X < 12) = 0.8488