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228 Chapter 4 Special Distributions Fitting the Poisson Distribution to Data Poisson data invariably refer to the numbers of times a certain event occurs during each of a series of âunitsâ (often time or space). For example, X might be the weekly number of trafï¬c accidents reported at a given intersection. If such records are kept for an entire year, the resulting data would be the sample k1 , k2 , . . . , k52 , where each ki is a nonnegative integer. Whether or not a set of ki âs can be viewed as Poisson data depends on whether the proportions of 0âs, 1âs, 2âs, and so on, in the sample are numerically similar to the probabilities that X = 0, 1, 2, and so on, as predicted by p X (k) = eâÎ» Î»k /k!. The next two case studies show data sets where the variability in the observed ki âs is consistent with the probabilities predicted by the Poisson distribution. Notice in each case that n the Î» in p X (k) is replaced by the sample mean of the ki âsâthat is, by kÌ = (1/n) ki . c=1 Why these phenomena are described by the Poisson distribution will be discussed later in this section; why Î» is replaced by kÌ will be explained in Chapter 5. Case Study 4.2.2 Among the early research projects investigating the nature of radiation was a 1910 study of Î±-particle emission by Ernest Rutherford and Hans Geiger (152). For each of 2608 eighth-minute intervals, the two physicists recorded the number of Î± particles emitted from a polonium source (as detected by what would eventually be called a Geiger counter). The numbers and proportions of times that k such particles were detected in a given eighth-minute (k = 0, 1, 2, . . .) are detailed in the ï¬rst three columns of Table 4.2.3. Two Î± particles, for example, were detected in each of 383 eighth-minute intervals, meaning that X = 2 was the observation recorded 15% (= 383/2608 Ã 100) of the time. Table 4.2.3 No. Detected, k Frequency Proportion p X (k) = eâ3.87 (3.87)k /k! 0 1 2 3 4 5 6 7 8 9 10 11+ 57 203 383 525 532 408 273 139 45 27 10 6 2608 0.02 0.08 0.15 0.20 0.20 0.16 0.10 0.05 0.02 0.01 0.00 0.00 1.0 0.02 0.08 0.16 0.20 0.20 0.15 0.10 0.05 0.03 0.01 0.00 0.00 1.0 (Continued on next page)