X =Number of "hits" in a fixed interval of
space or time (interval can be area
or volume)
The picture can model: X, the distribution of hits in the big
square shown
(with Lambda = 2.5 and X = 4 for this result--as
shown. P(X=4)= .1336),
Or Y, the distribution of hits in one little square, with 25
replications shown: Then Y = 1 for 4 squares, and Y = 0 for 21 squares.
Y = 2, 3, 4,....for 0 squares. Here Lambda = expected number for
one square, which = .10.
V-2 Rocket hits on London in World War II - A Real Application! *
During World War II, London was assaulted with German flying-bombs on V-2 rockets. The British were interested in whether or not the Germans could actually target their bomb hits or were limited to random hits with their flying-bombs. R. D. Clarke in his article, An Application of the Poisson Distribution, which appeared in the JOURNAL OF THE INSTITUTE OF ACTUARIES Vol 72 (1946), p. 481, shows the analysis which led the British to determine whether or not the Germans could target their bombs or were merely limited to random hits.
Before we turn the analysis over to YOU, it should be noted that this analysis is very important. For if the Germans could only randomly hit targets, then deployment throughout the countryside of various security installations would serve quite well to protect them, as random bombing over a wide range was unlikely to hit a given target. However, if the Germans could actually target their flying-bombs, then the British were faced with a more potent opponent and deployment of security installations would do little to protect them.
The British mapped off the central 24 km
by 24 km region of London into 1/2 km by 1/2 km square areas [The
town really isn't square; they used 576 little square areas of the big
square.]. Then they recorded the number of bomb hits in each of these
areas, noting their location, and this data is in the table below:
| # of bomb hits per (1/4 km2) area (k) | 0 | 1 | 2 | 3 | 4 | > 5 (actual
value was 7) |
Total |
| # (1/4 km2)of areas with k bomb hits | 229 | 211 | 93 | 35 | 7 | 1 | 576 areas
[check it] |
| Number of bombs for these areas | 0 | ? | 186 | ? | ? | ? | 537 |
| Assuming Poisson, P(Y=k) | e-lambda = ? | ? | ? | ? | ? | ? | 1 |
| Proportion of areas with k bomb hits
(# with k / total #of areas.) |
? | ? | ? | ? | ? | ? | 1 |
A) Did the Germans have capability of targeting
or were the V2 rockets falling randomly on London?
Analyze as follows: ( fill in the table, answer
the questions.)
a) How many bombs were dropped total?
229 areas got 0: contributes 0 x
229; 211 areas got 1 : contributes 1 x 211; 93 areas
got 2: contributes 2 x 93, etc, multiply and sum.
b) IF the bombs are falling randomly, the number Y of bombs to hit any particular small area should have a Poisson distribution. What is Lambda for one small area? The average number of bombs we would expect to hit such an area. We can estimate this using the data--the average number for each single small square area is approximately the total number of bombs divided by the total number of squares. Find this estimate of Lambda. (3 decimal places).
c) The number Y of bombs to hit a small area should have a Poisson distribution with the Lambda you found. Calculate the values P(Y =k) for k = 0, 1, 2, 3, 4, and find P(Y > 5) by subtracting the sum of the others from 1.
d) Find the actual proportions from the data for k = 0, 1, 2, 3, 4, > 5.
e) Compare the actual proportions with the model Poisson proportions. What do you think: is Poisson a reasonable model for the data?
B) Often in spatial data, or demographic spatial
data, clusters appear. Are they by chance, or are they evidence
of some "cause" (e.g. the Germans could "aim")? These are especially
"hot" issues when they are clusters of certain types of cancer, for instance;
and the public resists the idea that they might have happened by chance.
Use Siegrist's Two-Dimensional
Poisson applet to explore this:
a) The applet will only allow an approximation
to the bombs. Use w = 1 and r = 1 to see approximately what happens
in one square area, if you expect 1 bomb per square on average. Do
a bunch of single runs and see the variability.
b) Now extend this to an area more like London.
With w = 5, the max here, (and r still = 1) you can see what the pattern
would be for a block of 25 square areas. Do a bunch of single runs
and note when the data looks evenly scattered and when it looks more clustered,
empty areas and fuller areas.
c) Here is another
simulation: (Results will appear after you put in a number and
hit Simulate.) They model 100 "blocks", and the number to put in is 100--if
you expect 1 bomb per square, you expect 100 in 100 squares.
Run a few times and think of a Londoner coming out of the bomb shelters
to see what was hit on the block this time... (Here you can refine;
take your Lambda from part Ab, multiply by 100, round to a whole
number, to get even a closer simulation.)
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*(Writeup adapted from LTC Mike
Johnson, US Military Academy, Math 391, Lesson 22 (http://www.dean.usma.edu/MATH/people/johnson_m/MA391_AY03-2/lesson_plans.htm).
I first saw it in Feller, William. 1966. Introduction to Probability
Theory: Volume I\/, Second Edition. New York: John Wiley and Sons. Chapter
XVII.5.)
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