Winkler in 1866 extended Gauss's inequality to
rth moments where
r > 0 and the distribution is unimodal with a mode of zero. This is sometimes called Camp–Meidell's inequality. : P( | X | \ge k ) \le \left( \frac{ r } { r + 1 } \right)^r \frac{ \operatorname{ E }( | X | )^r } { k^r } \quad \text{if} \quad k^r \ge \frac{ r^r } { ( r + 1 )^{ r + 1 } } \operatorname{ E }( | X |^r ), : P( | X | \ge k) \le \left( 1 - \left[ \frac{ k^r }{ ( r + 1 ) \operatorname{ E }( | X | )^r } \right]^{ 1 / r } \right) \quad \text{if} \quad k^r \le \frac{r^r} { (r + 1)^{r + 1} } \operatorname{E}( | X |^r ). Gauss's bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode due to the
Vysochanskiï–Petunin inequality. The latter has been extended by Dharmadhikari and Joag-Dev : P( | X | > k ) \le \max\left( \left[ \frac r {( r + 1 ) k } \right]^r E| X^r |, \frac s {( s - 1 ) k^r } E| X^r | - \frac 1 { s - 1 } \right) where
s is a constant satisfying both
s >
r + 1 and
s(
s −
r − 1) =
rr and
r > 0. It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions. ==See also==