Lagrange Distributions of the Second Kind and Weighted Distributions

Starting with the second Lagrange expansion, with $f( z )$ and $g( z )$ as two probability generating functions defined on nonnegative integers such that $g( 0 ) \ne 0$, we define and study a new class of discrete probability distributions called the Lagrange distributions of the second kind. This class has the probability function: \[ P( Y = y ) = \frac{( 1 - g'( 1 ) )}{y!}\left( \frac{d^y }{dz^y }( g( z ) )^y f( z ) \right)_{z = 0} \] for $y = 0,1,2, \cdots $ . Different families are generated by various choices of the functions $f( z )$ and $g( z )$. Families of the weighted distributions that correspond to the Lagrange distributions of the first kind are defined using different weight functions. For a particular form of the weight function, it is shown that, under some conditions, the weighted distributions of the members of the Lagrange distributions of the first kind belong to the class of the Lagrange distributions of the second kind. Weighted distributions of the Borel–Tanner distribution, Haight’...