gamma distribution

Contributed by: Somak Sengupta
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What is Gamma Distribution?

Gamma Distribution is a Continuous Probability Distribution that is widely used in different fields of science to model continuous variables that are always positive and have skewed distributions. It occurs naturally in the processes where the waiting times between events are relevant. 

Why do we need Gamma Distribution?

It is used to predict the wait time until the future events occur. As we shall see the parameterization below, the gamma distribution predicts the wait time until the k-th (Shape parameter) event occurs. 

The PDF of the Gamma Distribution

It is a two-parameter continuous probability distribution. Exponential distribution and Chi-squared distribution are two of the special cases which we’ll see how we can derive from the Gamma Distribution. The commonly used parameterization are as follows-

  1. Shape parameter = k and Scale parameter = θ.
  2. Shape parameter α = k and an Inverse Scale parameter β = 1/θ called a Rate parameter. In exponential distribution, we call it as λ (lambda, λ = 1/θ) which is known as the Rate of the Events happening that follows the Poisson process. While k is the number of events until which we are waiting for the expected event to occur.
  3. Shape parameter = k and a Mean parameter μ = k*θ = α/β.

The general formulation for the probability density function (PDF) is-

      —Eq. 1

where, the Gamma Function is defined as – Γ(α) = (α-1)! for all positive integers. The gamma function* is eventually derived from the following integral–

      —Eq. 2

*Note that Gamma Distribution and Gamma Function are two different concepts. 

Using the parameters as k (# of events and k>0) and θ (λ = 1/θ) where λ is the rate of the event, we can write the PDF (Eq. 1) of the Gamma Distribution as–

      —Eq. 3

where Γ(k) = (k-1)! 

Therefore, a random variable X is eventually denoted by- 

      —Eq. 4

Special Case of Gamma Distribution – Exponential Distribution (k=1) 

gamma distribution

Special Case of Gamma Distribution – Chi Squared Distribution (θ=2, k=n/2, n = degrees of freedom)

gamma distribution

Other Important things to know– Properties

Mean – kθ
Variance – kθ2
Skewness – 2/sqrt(k)

Few important points

  • Gamma distribution, Poisson’s Distribution, and Exponential Distribution models are different aspects of the same process — the Poisson process. 
  • Poisson distribution is used to model the number of events in the future(k)
  • On the other hand, Exponential distribution is used to predict the wait time until the very first event occurs(λ)
  • It is used to predict the wait time until the k-th event occurs.
  • The two parameters (k and θ) are both strictly positive. The reason is – k is the number of events (which can’t be negative) and λ[1/ θ] is the rate of events (again, can’t be negative).

If you found this helpful and wish to learn more such concepts, stay tuned for more blogs! Happy Learning.



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