Radioactive decay is a random process, of course. If we observe
decays over an interval of time that is very short compared with
the lifetime, the number of atoms stays constant to a good
approximation. The number of decays over that period of time, on
average, should be

(6) 
where we are approximating
. In practice,
because of fluctuations, there may be more or fewer decays. The
probability of observing decays is given by the Poisson
distribution:

(7) 
where is the factorial
. If we make
measurements of the decay over a total time period , then the
number of intervals in which exactly decays occur is given by

(8) 
Because of statistical fluctuations, this statement is true only on
average.