Consider the case of a nuclide  that decays into another  by some process  (emission of other particles, like electron neutrinos  and electrons e as in beta decay, are irrelevant in what follows). The decay of an unstable nucleus is entirely random in time so it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any instant in time. Therefore, given a sample of a particular radioisotope, the number of decay events  expected to occur in a small interval of time  is proportional to the number of atoms present , that is
Particular radionuclides decay at different rates, so each has its own decay constant . The expected decay  is proportional to an increment of time, :
The negative sign indicates that  decreases as time increases, as the decay events follow one after another. The solution to this first-order differential equation is the function:
where  is the value of  at time  = 0, with the decay constant expressed as 
We have for all time :
where  is the constant number of particles throughout the decay process, which is equal to the initial number of  nuclides since this is the initial substance.
If the number of non-decayed  nuclei is:
then the number of nuclei of  (i.e. the number of decayed  nuclei) is
The number of decays observed over a given interval obeys Poisson statistics. If the average number of decays is , the probability of a given number of decays  is