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a U. S. Ru ssi Sp ain nc e Fr a Br ita in 1776 Au str ia 232 Chapter 4 Special Distributions ary tion Austria 0 Britain France Spain 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 Russia u vol Re r a W 0 U.S. Figure 4.2.2 or more wars erupted on four separate occasions; the Poisson model (Column 4) predicted that no years would experience that many new wars. Most likely, those four years had a decided excess of new wars because of political alliances that led to a cascade of new wars being declared simultaneously. Those wars deï¬nitely violated the binomial assumption of independent trials, but they accounted for only a very small fraction of the entire data set. The other binomial assumptionâthat each trial has the same probability of successâholds fairly well. For the vast majority of years and the vast majority of countries, the probabilities of new wars will be very small and most likely similar. For almost every year, then, X i can be considered a binomial random variable based on a very large n and a very small p. That being the case, it follows by Theorem 4.2.1 that each X i , i = 1500, 1501, . . . , 1931 can be approximated by a Poisson distribution. One other assumption needs to be addressed. Knowing that X 1500 , X 1501 , . . . , X 1931 âindividuallyâare Poisson random variables does not guarantee that the distribution of all 432 X i âs will have a Poisson distribution. Only if the X i âs are independent observations having basically the same Poisson distributionâthat is, the same value for Î»âwill their overall distribution be Poisson. But Table 4.2.4 does have a Poisson distribution, implying that the set of X i âs does, in fact, behave like a random sample. Along with that sweeping conclusion, though, comes the realization that, as a species, our levels of belligerence at the national level [that is, the 432 values for Î» = E(X i )] have remained basically the same for the past ï¬ve hundred years. Whether that should be viewed as a reason for celebration or a cause for alarm is a question best left to historians, not statisticians. Calculating Poisson Probabilities Three formulas have appeared in connection with the Poisson distribution: k . 1. p X (k) = eânp (np) k! k 2. p X (k) = eâÎ» Î»k! 3. p X (k) = eâÎ»T (Î»Tk!) k The ï¬rst is the approximating Poisson limit, where the p X (k) on the left-hand side refers to the probability that a binomial random variable (with parameters n and p)