Introduction to Probability · Problem 9.1.9

Central Limit Theorem — Grinstead & Snell

Mathematics — Probability · central-limit-theorem-bernoulli-trials

Problem

Let SnS_n be the number of successes in nn Bernoulli trials with probability .8 for success on each trial. Let An=Sn/nA_n = S_n/n be the average number of successes. In each case give the value for the limit, and give a reason for your answer. (a) lim⁡n→∞P(An=.8)\lim_{n\to\infty} P(A_n = .8). (b) lim⁡n→∞P(.7n<Sn<.9n)\lim_{n\to\infty} P(.7n < S_n < .9n). (c) lim⁡n→∞P(Sn<.8n+.8n)\lim_{n\to\infty} P(S_n < .8n + .8\sqrt{n}). (d) lim⁡n→∞P(.79<An<.81)\lim_{n\to\infty} P(.79 < A_n < .81).

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