# 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.

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.