The development of traditional Lagrangian distributions (L class) requires some conditions on two analytic functions f(z) and g(z) : f(1) = g(1) = 1, g(0) ≠ 0 and both functions are infinitely differentiable with respect to z in [-1, 1]. This study asks the question: 'What happens if some of these conditions are not satisfied?' We examine the consequences of relaxing these conditions and define the class of generalized Lagrangian probability distributions (GL class), which requires only the following conditions: the existence of z0 > 0 such that f(z0) > 0, g(z0) > 0 and f( z) and g(z) are infinitely differentiable at z0. This dissertation gives various properties of this GL class, including the equivalence of the GL and the class of all nonnegative integer-valued discrete distributions, the relationship between L and GL classes, the limiting distributions of the GL class, moments and convolution properties of the GL class, some new discrete distributions derived from GL class and a bivariate distribution formed by one discrete and one continuous random variables. Some applications are given on the performance of a new discrete distribution, the quasi-negative binomial distribution.
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