Hi,
I am trying to solve this exercise.
X is real random variable,  E[X]=0, Var(X)=\sigma^2 then
for every k>0 holds:
P[X\geq k] \leq \frac{\sigma^2}{\sigma^2+k^2}
Prove.

I have used so far Chebyshev bound
Pr(|X|\geq c\sigma)\leq\frac{1}{c^2}
for c=\frac{k}{\sigma} it leads to
P[X\geq k]\leq \frac{\sigma^2}{a^2}-P[X\leq-k]
But I dont know what to do next. Do you have any ideas?

John