Basic Statistical Methods

Clopper-Pearson Exact Method

The formula for the Clopper-Pearson exact method uses the relationship between the Binomial distribution and Beta distribution to form the interval

$$[Beta(\alpha/2; X, n - X + 1), Beta(1-\alpha/2; X + 1, n - X)]$$

Given an observation, the lower $p$ and upper limits are given by

$$\sum_{k = X}^n {n \choose k} p_L^k (1-p_L)^{n-k} = \alpha/2; \quad \sum_{k = 0}^X {n \choose k} p_U^k (1-p_U)^{n-k} = \alpha/2.$$