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This thesis concerns discrete probability distributions with power-law tailbehavior. Discrete power-law distributions are reviewed together withtheir truncated versions. The Zipf distribution (Zipf, 1932) is then focusedon and a truncated version of the Zipf distribution is further studied. Thisdistribution is known as the discrete truncated power-law distribution(DTPL). The basic properties of the DTPL are derived and procedurefor estimating the parameters of the distribution are developed. Theusefulness of the distribution is shown through several data examples.