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The problem of random intervals on a line

Published online by Cambridge University Press:  24 October 2008

C. Domb
C. Domb
Affiliation:
Pembroke CollegeCambridge
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Extract

Suppose that events occur at random points on a line from t = −∞ to +∞, the probability of an event occurring between t and t + dt being λdt. If we select any interval of the line, say the interval [0, y], there will be a finite probability that it contains 0, 1, 2,…,r,…events; in fact, it is not difficult to show that these probabilities form a Poisson distribution, the probability that the interval contains r events being (see e.g. (1)). Consider the case when each event consists of an interval of length α (an event being characterized by its first point). What is the probability that the covered portion of the interval [0, y] lies between x and x + dx?

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 3 , July 1947 , pp. 329 - 341
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

REFERENCES

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