Gamma Distribution
The Gamma distribution was developed during the early stages of statistical development as the sampling distribution of several statistics. It has several applications in reliability engineering. It can be used to model the failure characteristics of standby systems. The Gamma distribution has:
Where t > 0, k 0, and λ > 0.
Here, 1/λ is a scale parameter, and k is sometimes called the index or shape parameter. Exponential, Erlang, and Chi-Square distributions are special cases of this distribution. I(k,λt) is the incomplete gamma function.
Similar to the Weibull distribution, this distribution can be expressed in several forms. If we substitute k = α and λ = 1 / β, we have:
If we substitute k = λ and λ = 1 / σ, we have:
If we substitute λt = e z and 1 / λ = eμ , we have the following form: