Some New Characterizations of Discrete Lagrangian Distributions

Summary

When one studies the characterization theorems for the binomial and Poisson probability distributions one begins to wonder whether these properties are characteristic of these two distributions only or does there exist a wider class of distributions having these as a sub-class, which possess these characterization properties. This paper provides an affirmative answer to this question by proving seven theorems which characterize the Lagrangian Poisson distribution given by

$$ P\left( {X = x} \right) = {\left( {1 + x\theta } \right)^{x - 1}}{M^x}{e^{ - M - xM\theta }}/x!,x = 0,1,2, \ldots ,M > 0,0 \leqslant \theta \leqslant {M^{ - 1}} $$

and the quasi-binomial distributions given by

$$ P\left( {X = x} \right) = \left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right)\frac{{p\pi }}{{1 + n\theta }}{\left( {\frac{{p + k\theta }}{{1 + n\theta }}} \right)^{k - 1}}{\left( {\frac{{\pi + \left( {n - k} \right)\theta }}{{1 + n\theta }}} \right)^{n - k - 1}}, $$

, 0<p<1, p+π = 1, 0≤θ< 1 and k = 0,l,2,…,n. Since θ = 0 gives the characterizations of the binomial and the Poisson distributions, the quasi-binomial and the Lagrangian Poisson distribution appear to provide the wider class of distributions where these characterizations are true.

Financial support of the National Research Council od Canada and NATO Scientific Affairs Division is gratefully acknowledged for this work.