4.4. Simulation sample-size determination. When designing a simulation study, it is important to determine how many times to sample the system behavior. Suppose it is desired to estimate the probability that some event pi occurs (e.g., maximum system response exceeds some critical value). The obvious estimator of pi is Pi = Ni n + Source: Markovic [1965]. VIL Estir yum likelihood expressicPROBABILISTIC MODELS AND OBSERVED DATA 504 in which N; is the observed number of occurrences of the event in a sample of size n. N: has a binomial distribution, but for large enough values of npi, N: is approximately normal. Use this approximation to (@) Find (1 - @) 100 percent two-sided confidence limits pi + ca on the true Pi given n and an observed value of the estimator pi. Replace p: by p: where necessary (6) Show that the necessary sample size to insure that the (1 - @) 100 percent confidence limits are within v100 percent of the true value of p: (that is, that ca { vpi) is n 2 ka/ 2 (1 - pi) where ka/2 is that value which a standardized normal random variable exceeds with probability @/2. The answer is a function of pi which is unknown before the experi- ment is performed. This implies that the engineer must estimate the value of Pi before the experiment. (c) In a rainfall generation study (Chow and Ramaseshan [1965]) it was desired to estimate p; (which was thought to be about 0.15) to within 100 = 20 percent of its true value with confidence (1 - @) 100 percent = 90 percent; find the required n