On bounding the absolute mean value

A well-known bound for the absolute value of the mean a function g ( X ) of a random variable X , | E ( g ( X ) ) | , is E ( g ( X ) ) 2 . Here, we use a sharper bound of Cauchy's inequality, proved by Hovenier (J. Math. Appl. 186 (1994) 156–160), to give upper bounds for the absolute of mean value, | E ( g ( X ) ) | . Some examples for particular distributions are also provided.