Gamma Distribution definition, formula and applications

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In probabilistic statistics, the gamma distribution is a two-parameter family of continuous probability distributions which is widely used in different sectors. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of it.

Gamma Distribution

A continuous random variable X is said to have a generalised gamma distribution with parameter α and β if its probability density function is defined as,

$f\left&space;(&space;x;\alpha,&space;\beta&space;\right&space;)=\frac{\beta&space;^{\alpha&space;}e^{-\beta&space;x}x^{\alpha&space;-1}}{\Gamma&space;\alpha&space;};x>&space;0,\alpha&space;>&space;0,\beta&space;>&space;0.$

Where α and β are two parameter and  α,β>0. Gamma distribution is a continuous probability distribution.

Properties

There are some important properties as following,

• Gamma distribution has two parameter  α and β.
• Mean of Gamma distribution (variate) is α/β.
• Variance of Gamma distribution (variate) is α/ β2.
• Characteristic function of gamma distribu­tion is .
• Moment generating function of gamma distribution is .
• The measure of skewness β1=.
• Measure of kurtosisβ2= . These measures show that gamma distribution is positively skewed and leptokurtic.
• If the value of  β=1; mean= variance, if  β>1; mean<variance and if  β<1; mean>variance.

Applications

According to wikipedia, there are some important applications,

• It has been used to model the size of insurance claims and rainfalls. This means that aggregate insurance claims and the amount of rainfall accumulated in a reservoir are modelled by a gamma process – much like the exponential distribution generates a Poisson process.
• The gamma distribution is also used to model errors in multi-level Poisson regression models, because a mixture of Poisson distributions with gamma distributed rates has a known closed form distribution, called negative binomial.
• In wireless communication, the gamma distribution is used to model the multi-path fading of signal power.
• In oncology, the age distribution of cancer incidence often follows the gamma distribution, whereas the shape and scale parameters predict, respectively, the number of driver events and the time interval between them.
• In neuroscience, the gamma distribution is often used to describe the distribution of inter-spike intervals.
• In bacterial gene expression, the copy number of a constitutively expressed protein often follows it, where the scale and shape parameter are, respectively, the mean number of bursts per cell cycle and the mean number of protein molecules produced by a single mRNA during its lifetime.
• In genomics, it was applied in peak calling step (i.e. in recognition of signal) in ChIP-chip and ChIP-seq data analysis.
• It is widely used as a conjugate prior in Bayesian statistics. It is the conjugate prior for the precision (i.e. inverse of the variance) of a normal distribution. It is also the conjugate prior for the exponential distribution.

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