in the sample and Bin(n; p) is a binomial random variable with n trials and probability of success p. Equivalently we can say that the Clopper–Pearson interval is with confidence level if is the infimum of those such that the following tests of hypothesis succeed with significance : H0: with HA: H0: with HA: . Because of a relationship between the binomial distribution and the beta distribution, the Clopper–Pearson interval is sometimes presented in an alternate format that uses quantiles from the beta distribution. where x is the number of successes, n is the number of trials, and B(p; v,w) is the pth quantile from a beta distribution with shape parameters v and w. Thus, , where: The binomial proportion confidence interval is then , as follows from the relation between the Binomial distribution cumulative distribution function and the regularized incomplete beta function. When is either or , closed-form expressions for the interval bounds are available: when the interval is and when it is . The beta distribution is, in turn, related to the F-distribution so a third formulation of the Clopper–Pearson interval can be written using F quantiles: where x is the number of successes, n is the number of trials, and F(c; d1, d2) is the c quantile from an F-distribution with d1 and d2 degrees of freedom. The Clopper–Pearson interval is an exact interval since it is based directly on the binomial distribution rather than any approximation to the binomial distribution. This interval never has less than the nominal coverage for any population proportion, but that means that it is usually conservative. For example, the true coverage rate of a 95% Clopper–Pearson interval may be well above 95%, depending on n and p. Thus the interval may be wider than it needs to be to achieve 95% confidence, and wider than other intervals. In contrast, it is worth noting that other confidence interval may have coverage levels that are lower than the nominal , i.e., the normal approximation (or "standard") interval, Wilson interval, Agresti–Coull interval, etc., with a nominal coverage of 95% may in fact cover less than 95%, even for large sample sizes. The definition of the Clopper–Pearson interval can also be modified to obtain exact confidence intervals for different distributions. For instance, it can also be applied to the case where the samples are drawn without replacement from a population of a known size, instead