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
CLASS PROJECT: (Practice using the Poisson distribution)
Project 1 LONDON BOMBS
The program bombs simulate the London Bombs example. One of these two
generators simulate V1 rockets with a very rudimentary guidance system.
One of these is uniform random (i.e. no better than blindly throwing darts)
Run both generators (Method A and Method B) and see if you can estimate
the parameter lambda for average number of hits per cell for each method.
Select 100 bombs or less to carry on each of two runs.
Based on the values for lambda you find, you should be able to estimate
the numbers of cells with 0,1,2,3 etc bomb hits. Decide which of the two
generators has a guidance system and why. Write up the results of your
report using any graphics you deem appropriate. You should comment on
any problems you ran into in data collection and discuss how you resolved
them. Be sure to discuss what you think the purpose of this exercise is and
how your report does or does not support the conclusions. You may work
in groups, if you wish, on this assignment and hand in one report.
* The data is taken from R.D. Clarke, An Application of the Poisson
Distribution, Journal of the Institute of Actuaries, vol. 72 (1946), p. 481.