A Monotonicity Property of a New Bernstein Type Operator

Abstract

In the present paper we prove that the probabilities of the P\'olya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (P\'olya urn distribution with no replacement). As a consequence, we show that the random variables with P\'olya urn distribution (with negative replacement) are stochastically ordered with respect to the parameter giving the initial distribution of the urn. An equivalent formulation of this result shows that the new Bernstein operator recently introduced in [3] is a monotone operator. The proofs are probabilistic in spirit and rely on various inequalities, some of which are of independent interest (e.g. a refined version of the reversed Cauchy-Bunyakovsky-Schwarz inequality).