3) Extra Credit Challenge: We have seen in class that the Poisson distribution with parameter A = np is a good approximation to the probability distribution of the number of successes in n independent trials when each trial has probability p of being a success, provided that n is large and p small. In fact, the Poisson distribution remains a good approximation even when the trials are not independent, provided that their dependence is weak. Furthermore, it is not required that each trial be identical. For example, if the i-th trial had a probability p; of being a success (i =1, 2,...,n), then the total number of successes that occur in the n trials is well approximated by a Poisson random variable with A = n E, Pi, if each Pi is "small" and each trial is independent or only "weakly dependent." This is called the Poisson paradigm, . Suppose an experiment has n identical and independent trials, with each trial having k possible outcomes with probabilities p1,..., Pk respectively. Using the Poisson paradigm, show that if all the p;'s are small then the probability that no trial outcome occurs more than once is approximately equal to exp(-n(n – 1) E=, P /2).

3) Extra Credit Challenge: We have seen in class that the Poisson distribution with parameter A = np is a good approximation to the probability distribution of the number of successes in n independent trials when each trial has probability p of being a success, provided that n is large and p small. In fact, the Poisson distribution remains a good approximation even when the trials are not independent, provided that their dependence is weak. Furthermore, it is not required that each trial be identical. For example, if the i-th trial had a probability p; of being a success (i =1, 2,...,n), then the total number of successes that occur in the n trials is well approximated by a Poisson random variable with A = n E, Pi, if each Pi is "small" and each trial is independent or only "weakly dependent." This is called the Poisson paradigm, . Suppose an experiment has n identical and independent trials, with each trial having k possible outcomes with probabilities p1,..., Pk respectively. Using the Poisson paradigm, show that if all the p;'s are small then the probability that no trial outcome occurs more than once is approximately equal to exp(-n(n – 1) E=, P /2).

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Counting And Probability

Section9.3: Binomial Probability

Problem 2E: If a binomial experiment has probability p success, then the probability of failure is...

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