Gamma Distribution
A continuous random variable $X$ is said to have an gamma distribution with parameters $\alpha$ and $\beta$ if its p.d.f. is given by
$$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{\alpha^\beta}{\Gamma(\beta)}x^{\beta -1}e^{-\alpha x}, & \hbox{$x>0;\alpha, \beta >0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
Mean of Gamma Distribution
The mean or expected value of gamma random variable is
$E(X)= \dfrac{\beta}{\alpha}$
Variance of Gamma distribution
The variance of gamma random variable is
$V(X) = \dfrac{\beta}{\alpha^2}$.
Harmonic Mean of Gamma Distribution
The harmonic mean of gamma random variable is
$H=\frac{\beta-1}{\alpha}$.
Mode of Gamma distribution
The mode of gamma random variable is
$\dfrac{\beta-1}{\alpha}$.
M.G.F. of Gamma Distribution
The moment generating function of gamma random variable is
