Most of the derivations of the Gamma distribution pdf I've seen on here use the Poisson distribution. My lecture notes use the Gamma distribution and the exponential inter-arrival time definition of a Poisson process to define the Poisson distribution in the first place, so I'm looking for an alternative derivation.
I've seen one other way of doing this, by stating the distribution and then inductively proving it's the sum of exponential random variables to prove its utility, but this seems backwards to me.
How could I derive the formula for the pdf of the Gamma distribution from scratch without using the formula for the Poisson distribution? For example, if someone burnt all of my textbooks and I couldn't remember the formula and needed to rederive it from easily-remembered properties like the sum of exponentials.