Special Case 1: Exponential Distribution

Gamma with 

The PDF of the exponential distribution is:

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Mean and Variance

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

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)

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