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Beta Distribution

This page defines the Beta-distribution formulas Quantum XL shares: the Clopper-Pearson exact interval for a proportion, and the order-statistic coverage used by nonparametric tolerance intervals.

Notation

Term Description
\(\mathrm{Beta}^{-1}(q;\ a,\ b)\) the \(q\)-quantile of the Beta distribution with shape parameters \(a\) and \(b\)
\(x\) number of successes (count)
\(n\) number of trials
\(\alpha\) significance level (\(1 -\) confidence)

Clopper-Pearson exact interval for a proportion

For \(x\) successes in \(n\) trials:

\[ \text{lower} = 1 - \mathrm{Beta}^{-1}\!\left(1 - \tfrac{\alpha}{2};\ n - x + 1,\ x\right) \qquad \text{upper} = 1 - \mathrm{Beta}^{-1}\!\left(\tfrac{\alpha}{2};\ n - x,\ x + 1\right) \]

with \(\text{lower} = 0\) when \(x = 0\) and \(\text{upper} = 1\) when \(x = n\).

Used by: Confidence Interval, the binomial-proportion interval.

Order-statistic coverage (nonparametric tolerance)

The fraction of a population captured between two order statistics follows a Beta distribution. The coverage that can be guaranteed at confidence \(C\) (with \(\alpha = 1 - C\)) is that distribution's \(\alpha\)-quantile:

\[ \text{coverage} \sim \mathrm{Beta}(a,\ b) \qquad P_{\text{guaranteed}} = \mathrm{Beta}^{-1}(\alpha;\ a,\ b) \]

Used by: Tolerance Interval, which gives the exact one- and two-sided values of \(a\) and \(b\).

See Also

References

  1. Clopper, C. J., and Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413.
  2. Krishnamoorthy, K., and Mathew, T. (2009). Statistical Tolerance Regions: Theory, Applications, and Computation. Wiley.