## Gamma Distributions

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# Gamma Distributions

The gamma distribution is categorised among continuous probability density distributions. A continuous random variable X with a shape parameter* α *and a scale parameter* θ* has a gamma distribution only if its probability density function is as follows;

The gamma distribution is used mostly to describe activities where there is relevance on the waiting times between events such as the arrival of a train.

# Special Cases

**Exponential Distribution:**To obtain the exponential distribution from the gamma distribution you only need to define the rate parameter as*β**=**1/θ*and set the shape parameter as*α=1*. Substituting these values into the gamma function generates the probability density function given below, which is basically the exponential distribution;

**Chi-squared Distribution:**To obtain the chi-squared distribution from the gamma distribution you only need to set the shape parameter as*α=**v/2*and the scale parameter as*θ=2*. Substituting this value into the gamma function generates the probability density function given below, which is basically the chi-squared distribution, denoted as*x*;^{2}

Where v is the degrees of freedom.

# Applications

Some of the industries where the gamma distribution is used include;

- Climatology: It is used to determine the amount of rainfall accrued in a basin.
- Financial services: It is used to determine the scope of loan defaults. It is also used to mass insurance claims.

- Queuing models: It is used to determine the stream of objects through manufacturing and distribution processes.