\(T\) is random variable which is the time between events where the number of events occur as Poisson distribution,
pdf: \(f\left ( t\right ) =\lambda e^{-\lambda t}\)
\(P\left ( T>t\right ) ={\displaystyle \int \limits _{t}^{\infty }} \lambda e^{-\lambda s}ds=e^{-\lambda t}\)
\(P\left ( T<t\right ) ={\displaystyle \int \limits _{0}^{t}} \lambda e^{-\lambda s}ds=-\left [ e^{-\lambda s}\right ] _{0}^{t}=-\left [ e^{-\lambda t}-1\right ] =1-e^{-\lambda t}\)
Probability that the waiting time for \(n\) events to occur \(\leq t\) is a GAMMA distribution. \(g_{n}\left ( t\right ) =\frac {\lambda }{\left ( n-1\right ) !}\left ( \lambda t\right ) ^{n-1}e^{-\lambda t}\)