**Contributed by: Somak Sengupta **

**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 future events occur. As we shall see the parameterization below, 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-

It is a two-parameter continuous probability distribution. Exponential distribution and Chi-squared distribution are two of the special cases, the derivation of which from Gamma Distribution we will see. The commonly used parameterizations are as follows-

- Shape parameter = k and Scale parameter = θ.
- 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. - Shape parameter = k and a Mean parameter μ = k*θ = α/β.

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

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–

*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–

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) **

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

**Other Important things to know– Properties**

Mean – kθ

Variance – kθ^{2}

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 guide helpful and want to learn more about such concepts, stay tuned!

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