distribution function for the number of events that might occur in the next N years is given by the well-known binomial distribution, with expected value p*N, but with some chance for more than this number of events occurring during the next N years, and some chance of less. For example, if p* = 0.15 (a 15 percent chance of an event occurring each year) and N = 20 years, the expected number of events over the 20-year period is 0.15 ×20 = 3 events. We can also calculate the standard deviation for this amount (= [p*(1- p*)N]1/2), which in this case is calculated to be 1.6 events. All this, however, assumes that we are certain that p* = 0.15. In most homeland security modeling, such certainty will not be possible because the assumptions here do not hold for terrorism events. A more sophisticated analysis is needed to show the implications of our uncertainty in p* in those cases.

A common model used to represent uncertainty in an event occurrence probability, p (e.g., a failure rate for a machine part), is the beta distribution. The beta distribution is characterized by two parameters that are directly related to the mean and standard deviation of the distribution of p; this distribution represents the uncertainty in p (i.e., the true value of p might be p*, but it might lower than p* or higher than p*). The event outcomes are then said to follow a beta-binomial model, where the “beta” part refers to the uncertainty and the “binomial” part refers to the variability. When the mean value of the beta distribution for p is equal to p*, the mean number of events in N years is the same as that calculated above for the simple binomial equation (with known p = p*). In our example, with mean p = p* = 0.15 and N = 20 years, the expected number of events in the 20-year period is still equal to 3. However, the standard deviation is larger. So, for example, if our uncertainty in p is characterized by a beta distribution with mean = 0.15 and standard deviation = 0.10 (a standard deviation nearly as great or greater than the mean is not uncommon for highly uncertain events such as those considered in homeland security applications), then the standard deviation of the number of events that could occur in the 20-year period is computed to be 2.5. This is 60 percent larger than the value computed above for the binomial case where p is assumed known (standard deviation of number of events in 20 years = 1.6), demonstrating the added uncertainty in future outcomes that can result from uncertainty in event probabilities. This added uncertainty is also illustrated in Figure A-1, comparing the assumed probability distribution functions for the uncertain p (top graph in Figure A-1) and the resulting probability distribution functions for the uncertain number of events occurring in a 20-year period (bottom graph in Figure A-1) for the simple binomial and the beta-binomial models. As indicated, the beta-binomial model results in a greater chance of 0 or 1 event occurring in 20 years, but also a greater chance of 7 or more events occurring, with significant probability up to and including 11 events. In this case, characterizing the uncertainty in the threat estimate is clearly critical when estimating the full uncertainty in future outcomes.

Proper recognition and characterization of both variability and uncertainty is important in all elements of a risk assessment, including effective interpreta-



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