Manuscript Oai265-B@2 · Oai265-B-v2.tex · section · 94b6f42f

Exact binomial alternatives

Exact binomial alternatives · lines 767–788

Oai265-B@2#eq:cpL, Oai265-B@2#eq:cpU

Binary pass records allow exact binomial inversion ClopperPearson.

For $0 < ζ < 1$ , $n \geq 1$ , define

ℓ (n, 0; ζ) = 0, u (n, n; ζ) = 1.

For $k > 0$ , $ℓ (n, k; ζ)$ solves

i = k \sum^{n} (i n) p^{i} (1 - p)^{n - i} = ζ .

For $k < n$ , $u (n, k; ζ)$ solves

i = 0 \sum^{k} (i n) p^{i} (1 - p)^{n - i} = ζ .

The relevant tails are strictly monotone on $0 < p < 1$ , so these solutions are unique.

Displayed formulas in this passage