A one-tailed version of Chebyshev's inequality is that for t>0 $P[X-E(X) \geq t] \leq \frac{1}{1 + t^2/Var(X)}$ i.e. $P[X-E(X)\geq t] \leq \frac{Var(X)}{Var(X) + t^2}$
I heard someone claimed that the inequality is also true for the other side $P[X-E(X)\leq t] \leq \frac{Var(X)}{Var(X) + t^2}$
I briefly looked through the proof of the one-tailed version of Chebyshev's inequality, and it doesn't seem to work for the other side's version. So I was wondering if the other side isn't true?
Has the other side version been studied before?
Thanks!