Sum over k of n Choose k by p^k by (1-p)^n-k by Absolute Value of k-np/Historical Note

Historical Note on Sum over $k$ of $\dbinom n k$ by $p^k$ by $\left({1 - p}\right)^{n-k}$ by $\left\lvert{k - n p}\right\rvert$

This result was stated by Abraham de Moivre in his $1730$ work Miscellanea Analytica for the case where $n p$ is an integer.

The general case was proved by Henri Poincaré in his Calcul des Probabilités of $1896$.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $68$