How well can we measure the asymptotic frequency? I have run a number of computer simulations...

 

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How well can we measure the asymptotic frequency? I have run a number of computer simulations in which a 'success' occurred with probability p. It would be no fun if I told you the value of p, but I will tell you the results of the simulations. Trials Successes 10 69 10 641 104 6353 105 63738 106 637102 107 6366524 1. Using Bayesian statistics, we showed that we can represent our knowledge of the value of p by a probability distribution (or probability density) P(p|n), which is the conditional probability density for p, given that an experiment of N trials produced n successes. P(pln) is found using Bayes' theorem. Note that P(np), the conditional probability of finding n successes in N trials when the probability is p, is just the binomial distribution that we have been studying. Find the location of the maximum of this probability density by taking the first derivative of P(p|n) and setting a -In P(pin) = 0. Find the general expression for the width of P(pln) by taking the second derivative. The procedure is essentially the same as the one we used for calculating the Gaussian approximation to the integrand in deriving Stirling's approximation. 2. Using the result you obtained above, what can you learn about the value of p from each of these trials? Is the information from the various trials consistent? (Suggestion: A spreadsheet can be very helpful for calculations in this problem.) How well can we measure the asymptotic frequency? I have run a number of computer simulations in which a 'success' occurred with probability p. It would be no fun if I told you the value of p, but I will tell you the results of the simulations. Trials Successes 10 69 10 641 104 6353 105 63738 106 637102 107 6366524 1. Using Bayesian statistics, we showed that we can represent our knowledge of the value of p by a probability distribution (or probability density) P(p|n), which is the conditional probability density for p, given that an experiment of N trials produced n successes. P(pln) is found using Bayes' theorem. Note that P(np), the conditional probability of finding n successes in N trials when the probability is p, is just the binomial distribution that we have been studying. Find the location of the maximum of this probability density by taking the first derivative of P(p|n) and setting a -In P(pin) = 0. Find the general expression for the width of P(pln) by taking the second derivative. The procedure is essentially the same as the one we used for calculating the Gaussian approximation to the integrand in deriving Stirling's approximation. 2. Using the result you obtained above, what can you learn about the value of p from each of these trials? Is the information from the various trials consistent? (Suggestion: A spreadsheet can be very helpful for calculations in this problem.)

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Solutions Step 1 1 Using Bayes theorem the conditional probability density for p given that an exper... View the full answer