Hi,

I am trying to solve this exercise.

X is real random variable, $\displaystyle E[X]=0, Var(X)=\sigma^2$ then

for every k>0 holds:

$\displaystyle P[X\geq k] \leq \frac{\sigma^2}{\sigma^2+k^2}$

Prove.

I have used so far Chebyshev bound

$\displaystyle Pr(|X|\geq c\sigma)\leq\frac{1}{c^2}$

for $\displaystyle c=\frac{k}{\sigma}$ it leads to

$\displaystyle P[X\geq k]\leq \frac{\sigma^2}{a^2}-P[X\leq-k]$

But I dont know what to do next. Do you have any ideas?

John