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Distribution Of The Sum Of Independent Unity-Truncated Logarithmic Series Variables, Russell James Wayland

Dissertations and Theses

Let X₁, X₂, ..., X_n be n independent and identically distributed random variables having the unity-truncated logarithmic series distribution with probability function given by f(x;θ) = αθ^X/x x ε T where α = [-log(1-θ) - θ]^-1, 0 < θ < 1, and T = {2,3,...,∞}. Define their sum as Z = X₁ + X₂ + ... + X_n.

We derive here the distribution of Z, denoted by p(z;n,θ), using the inversion formula for characteristic functions, in an explicit form in terms of a linear combination of Stirling numbers of the first kind. A recurrence relation for the probability function p(z;n,θ) is obtained …