I want to prove the following result: Let $X$ be a nonnegative random variable defined on a probability space $\Omega$. Then
$\sum_{\omega\in\Omega:\ X(\omega)\geq\frac{\mathbb{E}X}{2}}\mathbb{P}(\omega)X(\omega) \geq \frac{\mathbb{E}X}{2}$
I need it in another proof, but I am not sure whether it is true at all. Can anyone help me with the proof? Thanks