Gamma distribution

Gamma distribution is a two-parameter family of continuous probability distributions. It is frequently used in both the natural and social sciences as a simple model of the times to events in a Poisson process.

Definition

The gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter., the symbol Γ(n) is the gamma function and is defined as (n−1)! :

The variable X is said to be gamma-distributed with the shape k and the scale θ.

Properties

Probability density function

The probability density function (pdf) of the gamma distribution shows the likelihood of a random variable X, given the parameters α and β.

Cumulative distribution function

The cumulative distribution function (cdf) of a random variable X gamma-distributed is the probability that X will take a value less than or equal to x.

Characteristic function

The characteristic function of a gamma-distributed random variable X with parameters α and β is derived by taking the inverse Fourier transform of the characteristic function.

Applications

The gamma distribution has been used to model various types of processes in the fields of life testing, reliability engineering, queuing theory, biology, medical sciences, hydrology, and many others.

See also

• Exponential distribution
• Beta distribution
• Chi-squared distribution
• Erlang distribution

References

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