In 1949, M. Kac defined a statistic which appears useful for statistical problems arising in insurance, biology, and telephone engineering . In those fields, the natural observation period is a fixed time period during which a random number of observations would be obtained. The distribution of the number of observations is given by a Poisson distribution. Distributions of this statistic can be used to determine upper and lower confidence contours for an unknown distribution, or in testing a distribution hypothesis. The purpose of this note is to extend the authors' earlier results, , to the two-sided Kac statistic.
"Distributions of a M. Kac Statistic." Ann. Math. Statist. 38 (6) 1919 - 1923, December, 1967. https://doi.org/10.1214/aoms/1177698629