Suppose that $X$ measures the half-life of a radioactive element, with decay rate $\lambda$ (per unit of time).
Starting from a population of $N$ particle, I believe you can model the number of particles that decay over a unit of time as a $Poisson(N\lambda)$, and the time until the first decay is detected is modeled using an $Exponential(N\lambda)$.
My question is: How can we model the distribution of the half-life $X$, i.e., starting from a population of $N$ particles, the amount of time it takes for $N/2$ of the particles to have decayed ?
Assuming the decays are independent from one another, it seems an Erlang distribution could be appropriate https://en.wikipedia.org/wiki/Erlang_distribution? Should I then use $f(x,N/2,N\lambda)$ (where the notation refers to the Wikipedia article)? If the half-life intuition is correct, it would seem $f(x,N/2,N\lambda)$ should then be invariant to the choice of $N$, but that's not obvious to me looking at the algebraic for of the Erlang's pdf. Is that correct? If so, there should be a single distribution $g = f(x,N/2,N\lambda)$ for all $N$ that models half-lifes?