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.
PDF of Gamma Distribution
A continuous random variable X is said to have a generalized gamma distribution with parameter α and β if its probability density function is defined as,
![\[ f\left ( x;\alpha ,\beta \right )=\frac{\beta ^{\alpha }e^{-\beta x}x^{\alpha -1}}{\Gamma \alpha }\ ; x> 0,\alpha > 0,\beta > 0 \]](https://www.statisticalaid.com/wp-content/ql-cache/quicklatex.com-c1574cfedbff86a36c9dea44610cf874_l3.png "Rendered by QuickLaTeX.com")
Where α and β are two parameter and α,β>0. Gamma distribution is a continuous probability distribution.
Example
Any multistep process in which each step takes the same amount of time. In molecular rearrangements, this is a regular occurrence. The time it takes for three phosphorylation events to occur, for example, is Gamma distributed if all phosphorylation events occur at the same rate.
Origin/History
According to Johnson et al. (1994, p. 343), the genesis of the gamma distribution may be traced back to Laplace (1836), who defined it as the distribution of a “precision constant.” Waiting times were modeled using the gamma distribution. In life testing, for example, the time it takes to reach “death” is a random variable with a gamma distribution (Hogg et al. 2013, p. 156). The gamma distribution is used as a conjugate prior distribution for many scale parameters in Bayesian statistics, such as the parameter in an exponential distribution or a normal distribution with a known mean.
When to Use Gamma Distribution
If you have a sharp lower bound of zero but no sharp upper bound, a single mode, and a positive skew, use the Gamma distribution with α > 1. In this situation, the Log Normal distribution is also an option. Gamma is particularly useful for encoding arrival times for groups of events. When you want to employ a bell-shaped curve for a positive-only quantity, a gamma distribution with a large value for α is also suitable.
Properties of Gamma Distribution
There are some important properties as following,
- It thas two parameter α and β.
- Mean of variate is α/β.
- Variance of variate is α/ β2.
- Characteristic function is .
- Moment generating function is .
- The measure of skewness β1=.
- Measure of kurtosis, β2= . 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.
- It 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, it 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.
Relations to Other Distribution
There are some relations to other distribution-
- The Gamma distribution is the continuous analog of the Negative Binomial distribution.
- The special case where α=1 is an Exponential distribution.
- The special case where α=ν/2 and β=1/2 is a chi-square parametrized by ν.