Rutherford’s Contagious Distribution


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Rutherford’s contagious distribution (or simply the Rutherford distribution) was inspired by the Pólya urn model, from which it arises naturally [1]. The distribution, built on prior work by Woodbury [2] concerns the probability of a success at any trial which depends linearly on the number of previous successes.

Woodbury considered a general Bernoulli scheme where the probability of a success depends on the number of previous successes, formulating the equation

P(n + 1, x + 1) = pxP(n, x) + (1- p x+1) P (n, x + 1).

Where

  • px = probability of success after x previous successes,
  • P(n, x) = probability of x successes in n trials.

If no pairs of px’s are equal, then the following formula can be obtained

Rutherford’s Contagious Distribution Formula

Rutherford’s contagious distribution detailed a special case of the formula. The idea is when a white ball is drawn from the urn, it is replaced with α other balls. This leads to a clustering of secondary cases around the first ball drawn. Rutherford used the linear function where px is determined by just two parameters:

px = p + cx (c > 0),

implying that

  • n < q/α if α > 0, and
  • n < –p/α if α < 0.

Rutherford’s special case formula avoids product notation:

rutherford's contagious distribution

Note, the distribution was proposed by R.S.G. Rutherford; there is no connection to Ernest Rutherford’s distribution that describes the scattering of alpha particles in physics.

References

[1] Rutherford, R. S. G. (1954). On a Contagious Distribution. The Annals of Mathematical Statistics, 25(4), 703–713. http://www.jstor.org/stable/2236654

[2] Woodbury, M. (1949). On a probability distribution. The Annals of Mathematical Statistics, 20, pp. 311-313.