de Moivre-Laplace Theorem

The sum of those terms of the Binomial Series of $(p+q)^s$ for which the number of successes $x$ falls between $d_1$ and $d_2$ is approximately

$$Q\approx {1\over\sqrt{2\pi}} \int_{t_1}^{t_2} e^{-t^2/2}\,dt,$$ (1)

where

$$t_1 \equiv {d_1-{\textstyle{1\over 2}}-sp\over\sigma}$$ (2)

$$t_2 \equiv {d_2+{\textstyle{1\over 2}}s-sp\over\sigma}$$ (3)

$$\sigma \equiv \sqrt{spq}.$$ (4)

Uspensky (1937) has shown that

$$Q={1\over\sqrt{2\pi}}\int_{t_1}^{t_2}e^{-t^2/2}\,dt+{q-p\ove... ...t[{(1-t^2){1\over 2\pi} e^{-t^2/2}}\right]^{t_2}_{t_1}+\Omega,$$ (5)

where

$$\vert\Omega\vert < {0.12+0.18\vert p-q\vert\over \sigma^2}+e^{-3\sigma/2}$$ (6)

for $\sigma\geq 5$.

A Corollary states that the probability that $x$ successes in $s$ trials will differ from the expected value $sp$ by more than $d$ is

$$P_\delta \approx 1-2\int_0^\delta \phi(t)\,dt,$$ (7)

where

$$\delta\equiv {d+{\textstyle{1\over 2}}\over\sigma}.$$ (8)

Uspensky (1937) showed that

$$Q_{\delta_1} \equiv P(\vert x-sp\vert\leq d)$$

$$= 2\int_0^{\delta_1} \phi(t)\,dt+{1-\theta_1-\theta_2\over\sigma}\phi(\delta_1)+\Omega_1,$$ (9)

where

$$\delta_1 \equiv {d\over\delta}$$ (10)

$$\theta_1 \equiv (sq+d)-\lfloor sq+d\rfloor$$ (11)

$$\theta_2 \equiv (sp+d)-\lfloor sp+d\rfloor$$ (12)

and

$$\vert\Omega_1\vert < {0.20+0.25\vert p-q\vert\over\sigma^2} +e^{-3\sigma/2},$$ (13)

for $\sigma\geq 5$.

References

Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, 1937.