Computer Codes for Exact Decision Levels and Errors of the First Kind When the Blank Count Time Is an Integer in [1, 20] Times Greater than the Sample Count Time
W.E. Potter (Consultant)
In the past exact computations utilizing modified Bessel functions of integral order have been discussed for decision levels and the associated errors of the first kind in paired counting when the blank count is Poisson distributed with a known expected value. This paper transforms the net count to an integer and assumes that the blank count is Poisson distributed with known expected value. Utilizing the Poisson probability density function, a function in C++ is written to compute the exact probability density function for the transformed net count when it is less than zero, equal to zero and greater than zero. This function sums probabilities to determine the probability of obtaining a specific value for the transformed net count. The validity of the computation is then checked by summing probabilities over a wide range of transformed net counts and comparing to 1.0 and by computing the expected transformed net count and comparing to 0.0. The decision level is determined by summing the right tail of the probability density function and inverting from a transformed net count to a net count. Two separate codes are used; one uses double precision arithmetic and is applicable for smaller expected blank counts in the sample count time and the other uses long double precision arithmetic and is applicable for larger expected blank counts in the sample count time. The long double precision code takes substantially longer to execute. Because the codes have the ability to compute errors of the first kind, comparisons can be made with approximate solutions such as those of both Brodsky and Currie. The double precision code is found to be adequate for most applications and executes challenging problems in about three minutes on a newer home computer.
