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Probability Distribution Functions, Poisson's distribution

An example of applications of the Poisson distribution could be the counting of the number of α-particles emitted from a radioactive source in a given time interval. In the limit of n and for small probabilities y, the binomial distribution approaches the Poisson distribution. Setting λ=ny, with y the probability for an event in the binomial distribution we can show that limn(nx)yx(1y)nxeλ=x=1λxx!eλ.