Poisson Trials

A number Trials in which the probability of success varies from trial to trial. Let be the number of successes, then

$\begin{displaymath} \mathop{\rm var}\nolimits (x)=spq-s{\sigma_p}^2, \end{displaymath}$

(1)

where ${\sigma_p}^2$ is the Variance of

and $q\equiv(1-p)$ . Uspensky has shown that

$\begin{displaymath} P(s,x)=\beta {m^xe^{-m}\over x!}, \end{displaymath}$

(2)

where

$\displaystyle \beta$	$\textstyle =$	$\displaystyle [1-\theta g(x)]e^{h(x)}$	(3)
$\displaystyle g(x)$	$\textstyle =$	$\displaystyle {(s-x)m^3\over 3(s-m)^3}+{x^3\over 2s(s-x)}$	(4)
$\displaystyle h(x)$	$\textstyle =$	$\displaystyle {mx\over s}-{m^2\over 2s^2}(s-x)-{x(x-1)\over 2s}$
	$\textstyle =$	$\displaystyle p\left[{{x\over 2}\left({1+{1\over m}}\right)-{(x-m)^2\over 2m}}\right]$	(5)

and $\theta\in(0,1)$ . The probability that the number of successes is at least

is given by

$\begin{displaymath} Q_m(x)=\sum_{r=x}^\infty {m^re^{-m}\over r!}. \end{displaymath}$

(6)

Uspensky gives the true probability that there are at least

successes in

trials as

$\begin{displaymath} P_{ms}(x)=Q_m(x)+\Delta, \end{displaymath}$

(7)

where

$\displaystyle \vert\Delta\vert$	$\textstyle <$	$\displaystyle \left\{\begin{array}{ll} (e^\chi-1)Q_m(x+1) & \mbox{for $Q_m(x+1)... ...-Q_m(x+1)] & \mbox{for $Q_m(x+1)\leq {\textstyle{1\over 2}}$}\end{array}\right.$
			(8)
$\displaystyle \chi$	$\textstyle =$	$\displaystyle {m+{\textstyle{1\over 4}}+{m^3\over s}\over 2(s-m)}.$	(9)