An Extremal Problem about Probability Distributions

A given probability distribution density is multiplied by all positive functions with a fixed ratio of upper and lower bounds. The products are normed so as to obtain probability densities again. The value of the variance in the class of probability distributions obtained by this sort of modification of the given distribution is studied. It is shown that the upper bound of the variance is attained for a piecewise constant modifying function shaped as a “rectangular trough”. A similar statement holds for the minimal variance. It is shown that the distribution with maximal variance is unique in the class in question.