Geodesic
Distribution of the Sum of Variates from Truncated Discrete Populations
Canadian mathematical bulletin, Tome 15 (1972) no. 3, pp. 395-398

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Distributions often arise in practice where one or more values of the variate are unobserved. Various practical problems have been referred by Finney [1], David and Johnson [4], Bliss and Fisher [3]. It is of interest to know the exact distribution of the sum of variâtes from such truncated discrete population. In this paper, utilising the property of characteristic function certain general results are shown. The distribution of the sum of independent variables from a discrete population, truncated by any set of s distinct values, follows from them immediately. Using these results the exact distributions of the sum, from binomial, poisson, negative binomial and geometric population, truncated from anywhere, are derived. 
Saleh, A. K. MD. Ehsanes; Rahim, M. A. Distribution of the Sum of Variates from Truncated Discrete Populations. Canadian mathematical bulletin, Tome 15 (1972) no. 3, pp. 395-398. doi: 10.4153/CMB-1972-072-1
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     title = {Distribution of the {Sum} of {Variates} from {Truncated} {Discrete} {Populations}},
     journal = {Canadian mathematical bulletin},
     pages = {395--398},
     year = {1972},
     volume = {15},
     number = {3},
     doi = {10.4153/CMB-1972-072-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-072-1/}
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