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Variables whose distributions achieve the boundary value of Chebyshev's inequality are characterized and it is found that non-constant variables with this property are symmetric discrete with at most three values. Nevertheless, the bound of Chebyshev's inequality remains optimal for the class of continuous variables.doi:10.3844/jmssp.2010.47.51 fatcat:p7g6ndq57zf4zeak4s6lv6ufau