# Gamma Distribution definition, formula and applications

### Gamma Distribution

A continuous random variable X is said to have a generalised gamma distribution with parameter α and β if its probability density fumction 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 of Gamma distribution

• 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.