I1 n

and now we will analyze each of the terms on the right hand side. First, N(N — 1)(N — 2) ... (N — k + 1), were we to expand it out would be a polynomial in N, that is it would take the form Nk + c1Nk — 1 + ..., so that the first fraction on the right hand side approaches 1 as N increases. The second fraction is independent of N. As N increases, the denominator of the third fraction approaches 1, and the numerator, as you recall from Chapter 2, the limit as Nof [1 — (1 /N)]N is exp(— 1). We thus conclude that in the limit of large N, small p with their product constant, the binomial distribution is approximated by the Poisson with parameter l = Np (for which we set t = 1 implicitly).

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