Use of Lagrange Expansion for Generating Discrete Generalized Probability Distributions

Abstract

Considering $g( t )$ and $f( t )$ as two probability generating functions defined on nonnegative integers such that $g( 0 ) \ne 0$, we use Lagrange’s expansion, together with the transformation $t = u \cdot g( t )$, to define families of discrete generalized probability distributions by the name of Lagrange distributions as \[ \begin{gathered} \Pr \,[ {X = 0} ] = L( {g;f;0} ) = f( 0 ), \hfill \\ \Pr \,[ {X = x} ] = L( {g;f;x} ) = \frac{1}{{x!}}\frac{{d^{x - 1} }}{{dt^{x - 1} }}\{ {( {g( t )} )^x \cdot f'( t )} \}|_{t = 0} \hfill \\ \end{gathered} \]for $x = 1,2,3, \cdots $, where the different families are generated by assigning different values to $g( t )$ and $f( t )$. General formulas for writing down the central moments of Lagrange distributions are obtained and it is shown that they satisfy the convolution property. The double binomial family of Lagrange distributions is studied in greater detail as it gives a large number of discrete distributions, including Borel-Tanner distribution, Haight’s distribution, generalized Poisson and generalized negative binomial distributions, as particular cases.