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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 “units” (often time or space). For example, X might be the weekly
number of traffic 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 ’s can be viewed as Poisson data depends on whether
the proportions of 0’s, 1’s, 2’s, 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) = e−λ λk /k!. The next
two case studies show data sets where the variability in the observed ki ’s is consistent
with the probabilities predicted by the Poisson distribution. Notice in each case that
n
the λ in p X (k) is replaced by the sample mean of the ki ’s—that is, by k̄ = (1/n) ki .
c=1
Why these phenomena are described by the Poisson distribution will be discussed
later in this section; why λ is replaced by k̄ 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 α-particle emission by Ernest Rutherford and Hans Geiger (152).
For each of 2608 eighth-minute intervals, the two physicists recorded the number of α 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 first three columns of Table 4.2.3. Two α particles, for example,
were detected in each of 383 eighth-minute intervals, meaning that X = 2 was
the observation recorded 15% (= 383/2608 × 100) of the time.
Table 4.2.3
No. Detected, k
Frequency
Proportion
p X (k) = e−3.87 (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)