< Back to Probability Distribution List

**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) = *p _{x}P*(

*n*,

*x*) + (1-

*p*

_{x}_{+1})

*P*(

*n*,

*x*+ 1).

Where

*p*= probability of success after_{x }*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 *p _{x} *is determined by just two parameters:

*p _{x}*

_{ }**=**

*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:

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.