About how people come up with Gamma distribution

Question

The gamma distribution is given by $f(x)= \frac{1}{\Gamma(\alpha) \theta^{\alpha}} x^{\alpha-1} e^{\frac{-x}{\theta}}$. I know this is a special type of Poisson distribution where it is counting the n th an event is happening rather than one. I know $\Gamma(\alpha)$ is $(\alpha-1)!$ Question is, how people derive that formula? Is there an intuitive way to remember that?

Answer 1

No, it is not at all a Poisson distribution. Poisson is a discrete distribution, this is continuous. In a Poisson process, the Poisson distribution counts the number of events that happen in a given period of time, the Gamma distribution tells you how long a time it takes for a given number of events to occur.

The connection between them is: if $X$ has a Poisson distribution with parameter $\lambda= t/\theta$ and $T$ has a Gamma distribution with parameters $\alpha$ and $\theta$ (where $\alpha$ is a positive integer), $\mathbb P(T \le t) = \mathbb P(X \ge \alpha)$. That is, the $\alpha$'th event happens by time $t$ if and only if there are at least $\alpha$ events up to time $t$.