Bounds on Negative Moments

Four sets of bounds on E(X- 1 ), E(X- 2 ) and var (X- 1 ) are computed in terms of μ = E(X),σ ² = E(X - μ )², and $a > 0$ for which P(X a) = 1. The four sets correspond to X unrestricted, X symmetric at μ, X symmetric and unimodal at μ, and X unimodal at m. For X unimodal, bounds on var (X- 1 ) are omitted since they appear to have no simple closed form.