Professor Rick Cornez
Course: Math 311 Probability

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As an example of a spatial distribution of random points consider the statistics of flying bomb hits in the south of London during World War II. The entire area is divided into a grid of N=576 small areas of size one quarter square kilometer each. The table below records the number of squares with 0, 1, 2, 3 etc hits each. The total number of hits is 537. The average number of hits per square is then 537/576=.9323 hits per square. It can be shown that if the targeting is completely random, then the probability that a square is hit with 0,1,2,3 etc hits is governed by a Poisson distribution .

Thus , the probability that a given square suffers k hits is :

where k=0,1,2,3 etc. In this case lambda is .9323.

The actual fit of the Poisson for the data is surprisingly good. It is interesting to note that most people believed in a tendency of the points of impact to cluster. If this were true there would be a higher frequency of areas with either no impacts or many impacts and a deficiency in the intermediate classes. The table indicates randomness and homogeneity of the area. We have here an instructive illustration of the fact that to the untrained eye randomness appears as regularity or tendency to cluster.

FLYING-BOMB HITS ON LONDON (576 cells with 537 hits)*

# of hits 0 1 2 3 4 5
# of cells with # of hits above 229 211 93 35 7 1
Poisson Fit 226.7 211.4 98.5 30.6 7.1 1.6