Consider a radioactive substance with concentration C, in atoms per litre, which is constant in space and time. The measured half-life is \(\tau\), in seconds, and the decay constant is \(\lambda = \ln(2)/\tau\). The number of decays per second per litre – the decay rate – is \(\lambda C\).
However, this is an average value, and the decay is a Poissonian process. The probability of \(k\) decays per litre in any given second-long period is:
The figure shows example Poisson distributions for \(\lambda C\) = 10, 50, 90 and 130.