Note on the probability distribution arising in the study of the Institute's examinations

Extract

The distribution discussed by Mayhew and Vajda (J.S.S. Vol. vi, 1946, pp. 67–75 (70)), is the binomial with negative index (having origin, i.e. first term, at n instead of o), about which a good deal has been written, and their problem is the one discussed by Yule in 1910 (J.R. Statist. Soc. Vol. LXXIII, p. 26). The authors base their proof on an idea of Laplace, but the full power of the characteristic function derivation can be seen if we go further than they did. The results are of some interest, particularly as Kendall (Advanćed Theory of Statistics, Vol. 1, §5.13) does not go beyond deriving mean and variance by the elementary method in this case.