Poisson Approximation of Binomial Probabilities

^{T}P_{(k out of n)} =  (e^{—np})(np^{k}) k! 
e =  the base of the natural logarithms;  
n =  the number of opportunities for event x to occur;  
k =  the number of times that event x occurs or is stipulated to occur; and  
p =  the probability that event x will occur on any particular occasion; 
n  k  p  q 
binomial mean  
binomial SD  
discrepancy 
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The defining characteristic of a Poisson distribution is that its mean and variance are identical. In a binomial sampling distribution, this condition is approximated as p becomes very small, providing that n is relatively large. The mean and variance of a binomial sampling distribution are equal to np and npq, respectively (with q=1—p). As p approaches zero, the value of npq approaches that of np, and the binomial distribution accordingly approximates the form and properties of the Poisson.
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