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The binomial deriving method put lambda = np and finally the binomial p.m.f. become the Poisson one as n goes to infinity.

It seems that this is only proving that binomial distribution will approach the Poisson distribution as n goes to infinity, p goes to 0, and lambda stays constant, but it has nothing to do with deriving the p.m.f. of Poisson distribution.

So, the method has not solved my question that how does the p.m.f. of Poisson distribution come from.

I am doubtful for this.