Two-Sided Probability Bound for a Symmetric Unimodal Random Variable

For a symmetric unimodal random variable with known mode and variance, we construct a tight upper bound for the probability of falling outside an arbitrarily specified interval. We define the distribution on which the bound is achieved. In the special case when the distance between mode and the middle of the interval is less than 29% of its radius, the resulting bound is determined by the Gauss inequality. We have applied our result to the construction of robust confidence intervals.