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Many methods have been suggested to choose between distributions. There has been relatively less study to examine whether these methods accurately recover the distributions being studied. Hence, this research compares several popular distribution selection methods through a Monte Carlo simulation study and identifies which are robust for several types of discrete probability distributions. In addition, we study whether it matters that the distribution selection method does not accurately pick the correct probability distribution by calculating the expected distance, which is the amount of information lost for each distribution selection method compared to the generating probability distribution. [ABSTRACT FROM AUTHOR]