  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).
