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Poisson Trials

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

var(x)=spq-ssigma_p^2,
(1)

where sigma_p^2 is the variance of p_i and q=(1-p). Uspensky has shown that

P(s,x)=beta(m^xe^(-m))/(x!),
(2)

where

beta=[1-thetag(x)]e^(h(x))
(3)
g(x)=((s-x)m^3)/(3(s-m)^3)+(x^3)/(2s(s-x))
(4)
h(x)=(mx)/s-(m^2)/(2s^2)(s-x)-(x(x-1))/(2s)
(5)
=p[x/2(1+1/m)-((x-m)^2)/(2m)]
(6)

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))/(r!).
(7)

Uspensky gives the true probability that there are at least x successes in s trials as

P_(ms)(x)=Q_m(x)+Delta,
(8)

where

|Delta|<{(e^chi-1)Q_m(x+1) for Q_m(x+1)>=1/2; (e^chi-1)[1-Q_m(x+1)] for Q_m(x+1)<=1/2
(9)
chi=(m+1/4+(m^3)/s)/(2(s-m)).
(10)

See also

Trial

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Cite this as:

Weisstein, Eric W. "Poisson Trials." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoissonTrials.html

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