Applied probability and stochastic processes by Bryc W.

By Bryc W.

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The outcome X of such an experiment is a binomial random variable with unknown probability p of success, and we would estimate p  X=n. The trouble is that if we don't know how small the chances are, we might get none, estimating probability to be zero. Or we could run the experiment until we get the prescribed number of successes. The observation would then consist of a set of geometric random variables T1; : : : ; Tk with unknown mean ET = 1=p. We could then estimate p = 1=ET by taking the inverse of the arithmetic mean of Tj .

But before we do that, we should develop some intuition on the cases that can check the answers. Therefore we begin with simulation of probabilities or averages that are known. A Simulation of probabilities averages that are known should address the following questions. 1. How close the simulation answers are to the theoretical answers? Print them side-by-side. 2. How large the simulation should be? Is it worth to change simulation size from 1,000 to 10,000 trials? In order to answer this question, your simulation has to provide relative" rather than absolute answers.

1 See eg. W. Feller, An Introduction to Probability Theory, Vol II, Wiley, New York 1966. 3. 1 Probability generating functions For ZZ+-valued random variables it is convenient to consider the so called generating P p zk function Gz  = M ln z . 1i is elementary, as Gz = 1 k=0 k determines uniquely its Taylor series coe cients pk = PrX = k. 3 Characteristic functions The major nuisance in using the moement generating functions is the fact that the moment generating functions may not exist, when the de niting integral diverges.

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