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If X and Y are i.i.d Poisson variables with lambda1 and lambda2 then,

P (X = x | X + Y = n) ~ Bin(n, p) where p = lambda1 / lambda1 + lambda2

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The Poisson distribution is a limiting case of the binomial distribution when the number of trials is very large and the probability of success is very small. The Poisson distribution is used to model the number of occurrences of rare events in a fixed interval of time or space, while the binomial distribution is used to model the number of successful outcomes in a fixed number of trials.

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Q: How is poisson distribution related to binomial distribution?
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