# Poisson distribution

By 0x7df, Fri 18 September 2020, in category Uncategorized

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:

$$p(k) = \frac{\left[\lambda k\right]^k e^{-\left[\lambda C\right]}}{k!}$$

The figure shows example Poisson distributions for $$\lambda C$$ = 10, 50, 90 and 130.