Good morning,
I am trying to prove the following inequality:
$$ P(S \geq \mathbb{E}[S]+c) \leq \frac{\operatorname{Var}(S)}{\operatorname{Var}(S) + c²} $$ $$ \text{with } S = \sum_{k=1}^{n}{X_k} , X_k\geq 0 $$
I was trying to solve this inequality by the "generalized" version of the Chebyshev's inequality. Resulting in:
$$ P(S \geq \mathbb{E}[S]+c) \leq \frac{\mathbb{E}[S^2]}{(\mathbb{E}[S]+c)^2} = \frac{\operatorname{Var}(S) + \mathbb{E}[S]^2}{c^2 + \mathbb{E}[S]^2 + 2\mathbb{E}[S]c } $$
Best Answer
It can be shown by the Markov and not Cheby. inequality.