 
Summary: 1
Nonnegativity of Odd Functional Moments of
Positive Random Variables with Decreasing Density
Gerold ALSMEYER
Mathematisches Seminar, Universit¨at Kiel, D24098 Kiel, Germany
Abstract: In this note we give some results on the nonnegativity of odd
functional moments of random variables with a decreasing density, more
precisely we prove, by purely elementary arguments, E(X  EX) 0
for suitable functions that satisfy (x) = (x) for all x 0 and
random variables X 0 with a decreasing Lebesgue density on (0, ) or
counting density on IN0. The motivation came from a problem recently
published in Statistica Neerlandica, see Introduction below, concerning
a more specialized result.
Keywords: Functional moments, skewness, decreasing density.
1. Introduction and Results
Starting point and motivation for the present article has been Problem 234 in Statistica Neer
landica posed by R. Gill:
A positive, continuously distributed random variable X with finite mean µ and a
decreasing density f(x), x (0, ), is intuitively speaking skewed to the right; hence
its coefficient of skewness and more generally all its odd moments should be positive
