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Poisson Trials

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

\mathop{\rm var}\nolimits (x)=spq-s{\sigma_p}^2,
\end{displaymath} (1)

where ${\sigma_p}^2$ is the Variance of $p_i$ and $q\equiv(1-p)$. Uspensky has shown that
P(s,x)=\beta {m^xe^{-m}\over x!},
\end{displaymath} (2)

$\displaystyle \beta$ $\textstyle =$ $\displaystyle [1-\theta g(x)]e^{h(x)}$ (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 $x$ is given by
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 $x$ successes in $s$ trials as
\end{displaymath} (7)

$\displaystyle \vert\Delta\vert$ $\textstyle <$ $\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.$  
$\displaystyle \chi$ $\textstyle =$ $\displaystyle {m+{\textstyle{1\over 4}}+{m^3\over s}\over 2(s-m)}.$ (9)

© 1996-9 Eric W. Weisstein