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A recurrent theme of interest in probability and statistics is to determine the best bounds for two probabilities, Pr(X ≥ r) and Pr(s < X - μ < t), when only the mean μ and the standard deviation σ of a random variable X are known. This article addresses the issue under two circumstances, when X is arbitrary and when X is nonnegative. The answers are provided in a unified manner using only Markov's theorem. The existing literature on related inequalities is reviewed. Some examples are given to illustrate the use of the inequalities.
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B. K. Ghosh (2002) studied this question.
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