How do we know that radioactive decay is memoryless? Let $\tau$ be the random variable that describes the lifetime of a given particle. It seems to conform to common-sense that $\mathbf{P}(\tau>t+s|\tau>s)=\mathbf{P}(\tau>t)$, as it would be weird if the particle could ``keep track" of how long it has been around.
My question is, do we have any good reason to believe that $\tau$ is memoryless in this way, besides the common-sense reason I have provided? I would appreciate references.
 A: The experimental result that the rate of decay is measured to be proportional to the amount of undecayed particles actually confirms memorylessness.
This is because this result, together with the assumptions that the particles are not "communicating" with one another and that they are identical confirms that the probability density of the time to decay for each particle is exponential. If we assume no "communication" i.e. no interaction so that a particle's decay does not influence another's, then the times to decay are statistically independent variables, so that the rate of decay is proportional to $p(\tau)$, the pdf of the time to decay.
Once you observe this fact, you then need to know that the exponential distribution of time till decay is the unique probability distribution that can arise from an assumption of memorylessness. I discuss the proof of this fact in my answer here. Moreover, the argument works in reverse, so that not only does memorylessness imply exponentially distributed time till decay, but the latter also implies the former and they are logically equivalent.
