Suppose we wish to estimate the probability, PA, of some event, A . We do so by repeating an experiment n times and observing whether or not the event A occurs during each experiment. In particular, let

We then estimate PA using the sample mean of the Xi,

(a) Assuming is large enough so that the central limit theorem applies, find an expression for Pr (|Ṕ_A –P_A| < Ɛ). 

(b) Suppose we want to be 95% certain that our estimate is within ± 10% of the true value. That is, we want Pr (|Ṕ_A –P_A| < 0.1 P_A) = 0.95. How large does n need to be? In other words, how many times do we need to run the experiment?

(c) Let Y_n be the number of times that we observe the event during our n repetitions of the experiment. That is, let Y_n = X₁+ X₂+…+ X_n. Assuming that is chosen according to the results of part (b), find an expression for the average number of times the event is observed, P_{A »} 1) E [Y_n]. Show that for rare events (i. e.) is essential independent of P_A. Thus, even if we have no idea about the true value of P_A, we can run the experiment until we observe the event for a predetermined number of times and be assured of a certain degree of accuracy in our estimate of P_A.

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I , 0, I occured during ith experiment. otherwise