Estimates of Bounded Relative Error in Particle Counting

Abstract

A statistical problem arising in many fields of activity requires the estimation of the average number of events occurring per unit of a continuous variable, such as area or time. The underlying distribution of events is assumed to be Poisson; the constant to be estimated is the unknown parameter $\lambda$ of the distribution. A sampling procedure is proposed in which the continuous variable is observed until a fixed number $M$ of events occurs. Such a procedure enables us to form an estimate $l$, which with confidence coefficient $\alpha$ does not differ from $\lambda$ by more than 100 $\gamma$ per cent of $\lambda$. The values of $\gamma$ and $\alpha$ depend on $M$ but not on $\lambda$. Modifications of this procedure which are sequential in nature and have possible operational advantages are also described. These procedures are discussed in terms of a chemical problem of particle counting. It is clear, however, that they are generally applicable whenever the basic probability assumptions apply.