Groer P G
The University of Tennessee, Department of Nuclear Engineering, Knoxville 37996-2300, USA.
Radiat Prot Dosimetry. 2002;102(3):265-8. doi: 10.1093/oxfordjournals.rpd.a006095.
The stochastic fluctuations in the number of disintegrations, which had already been studied experimentally by Rutherford and other investigators at the beginning of the twentieth century, make estimation of net counting rates in the presence of background counts a challenging statistical problem. Exact and approximate Bayesian estimates of net count rates using Poisson and normal distributions for the number of counts detected during varying counting intervals are derived. The posterior densities for the net count rate are derived and plotted for uniform priors. The graphs for the exact, Poisson based, and for the approximate posterior densities of the background and net count rates, resulting from the normal approximation to the Poisson distribution, were compared. No practical differences were found when the number of observed gross counts is large. Small numerical differences in the posterior expectations and standard deviations of the counting rates appeared when the number of observed counts was small. A table showing some of these numerical differences for different background and gross counts is included. A normal approximation to the Poisson is satisfactory for the analysis of counting data when the number of observed counts is large. Some caution has to be exercised when the number of observed counts is small.
二十世纪初，卢瑟福和其他研究人员已通过实验研究了衰变次数的随机波动，这使得在存在本底计数的情况下估计净计数率成为一个具有挑战性的统计问题。本文推导了在不同计数间隔内检测到的计数次数使用泊松分布和正态分布时净计数率的精确和近似贝叶斯估计。推导了净计数率的后验密度，并针对均匀先验进行了绘图。比较了由泊松分布的正态近似得到的背景和净计数率的精确、基于泊松的以及近似后验密度的图形。当观察到的总计数数量很大时，未发现实际差异。当观察到的计数数量很小时，计数率的后验期望和标准差出现了小的数值差异。本文包含一个表格，展示了不同背景和总计数情况下的一些数值差异。当观察到的计数数量很大时，对泊松分布进行正态近似用于计数数据分析是令人满意的。当观察到的计数数量很小时，必须谨慎使用。
