  Chebyshev's inequality provides an upper bound on the tail probability of a
random variable based on its mean and variance. While tight, the inequality has
been criticized for only being attained by pathological distributions that
abuse the unboundedness of the underlying support and are not considered
realistic in many applications. We provide alternative tight lower and upper
bounds on the tail probability given a bounded support, mean and mean absolute
deviation of the random variable. We obtain these bounds as exact solutions to
semi-infinite linear programs. We leverage the bounds for distribution-free
analysis of the newsvendor model, monopolistic pricing, and stop-loss
reinsurance. We also exploit the bounds for safe approximations of sums of
correlated random variables, and to find convex reformulations of single and
joint ambiguous chance constraints that are ubiquitous in distributionally
robust optimization.
