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Binomial Distribution at Infinity

Consider a binomial distribution of XB(n,p)X\sim B(n,p).

It can be easily shown that P(X=k)=(nk)pk(1p)nkP(X=k)={n\choose k}p^k{(1-p)}^{n-k} for k=0,1,2,3,,nk=0,1,2,3,\ldots,n.

Now, let's take the limit of the above using nn \to \infty. Instead of having an infinitesimal pp, let's assume that it is given that npnp, the mean of the probability distribution function, is some finite value mm.

Find P(X=k)P(X=k) in terms of mm and kk for this new distribution, where k=0,1,2,3,k=0,1,2,3,\ldots, without looking anything up or reciting any formulas from memory.


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