Exact Confidence Intervals in Distributions with One Parameter

The Clopper–Pearson method of constructing a confidence interval for the probability of success in a binary population that follows a Bernoulli distribution is well known. This paper pedagogically justifies the Clopper–Pearson method and extends the method to all distributions with one parameter whose cumulative probability distribution functions are monotonically continuous with respect to their single parameter. The conservativeness of Clopper–Pearson confidence intervals is proved analytically. Clopper–Pearson confidence intervals are constructed for Poisson, geometric, non-central hyper-geometric, and exponential, etc. It turns out that such extensions result in various well-known confidence intervals in the literature.