Probability distribution for radioactive decay

Here is a rigorous justification for the statement in section 2.4 that the probability distribution for radioactive decay is found by substituting N(0) = 1 into the equation for the rate of decay. We know that the probability distribution must be of the form

where k is a constant that we need to determine. The atom is guaranteed to decay eventually, so normalization gives us

The integral is most easily evaluated by converting the function into an exponential with e as the base

which gives an integral of the familiar form R ecxdx = (1/c)ecx . We thus have

which gives the desired result: