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;

gamma distribution

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

  1. 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;


  1. 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 x2;


Where v is the degrees of freedom.

Applications

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

  1. Climatology: It is used to determine the amount of rainfall accrued in a basin.
  2. Financial services: It is used to determine the scope of loan defaults. It is also used to mass insurance claims.
  3. Queuing models: It is used to determine the stream of objects through manufacturing and distribution processes.

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