User:IssaRice/Gamma distribution

https://www.youtube.com/watch?v=Qjeswpm0cWY

ok, so given a poisson process, instead of asking the time until the next occurrence, we can ask for the total time taken until n things happen, and this will be the sum $T_{n}=\sum _{j=1}^{n}X_{j}$, where each $X_{j}\sim \mathrm {Exponential} (\lambda )$. Now this sum happens to have distribution $\mathrm {Gamma} (n,\lambda )$. This still doesn't explain what it means when $n$ (or $\alpha$ as this parameter is usually called) is not a positive integer.