Justification of the Poisson approximation property.

QUESTION:

Justification of the Poisson approximation property. Consider the PMF of a binomial random variable with parameters n and p. Show that asymptotically, as n ---+ 00, p ---+ O. while np is fixed at a given value >., this PMF approaches the P1-IF of a Poisson random variable with parameter >.. Solution. Using the equation >. = np, write the binomial PMF as I px (k) = (n _ n k ) ! k !p k(1 -ptk = n(n - l) ... (n- k+ l) . >.k . ( 1 _ ) n-k nk k ! n Fix k and let n --> 00. We have, for j = 1, . . . . k, n-k + j --> 1 . n Thus, for each fixed k, as n --> 00 we obtain