Moments

The next section deals with the Gamma distribution, which is a generalisation of the Exponential distribution. The moments of the exponential distribution can be found by setting $\alpha =1$ in the formulae for the Gamma distribution.

1. $𝖤\left[X\right]=\frac{1}{\beta }$,

2. $\mathrm{𝖵𝖺𝗋}\left[X\right]=\frac{1}{{\beta }^2}$.

Hence the expectation and standard deviation are the same, so the coefficient of variation is $1$ for all $\beta$. Note that the expectation decreases with $\beta$; the higher the rate of events the shorter the expected waiting time to the next event.