Enter the number of trials n, success probability p and upper bound x to get the cumulative probability P(X<=x) of a binomial distribution, for acceptance sampling, reliability and cumulative binomial analysis.
输入试验数 n、成功概率 p 与上限 x，求 P(X≤x)。
| Item | Formula / Note | Source / Standard |
|---|---|---|
| Binomial probability mass function | P(X = k) = C(n, k) · p^k · (1 - p)^(n - k) | Bernoulli J., Ars Conjectandi (1713) |
| Cumulative distribution function | F(k) = P(X <= k) = sum(i=0..k) C(n, i) · p^i · (1 - p)^(n - i) | NIST/SEMATECH e-Handbook of Statistical Methods §1.3.6.6 |
| Expectation and variance | E[X] = n·p; Var[X] = n·p·(1 - p) | NIST/SEMATECH e-Handbook of Statistical Methods §1.3.6.6 |
| Normal approximation condition | Usable when n·p >= 5 and n·(1 - p) >= 5, via N(np, np(1-p)) | NIST/SEMATECH e-Handbook of Statistical Methods, empirical rule |