Let $X$ be a non-negative random variable that is almost surely bounded but not identically zero, show that $$\displaystyle \mathop{\bf P}( X \geq \frac{1}{2} \mathop{\bf E} X) \geq \frac{1}{2} \frac{\mathop{\bf E} X}{\|X\|_\infty}.$$

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Exercise 1.45 (Basic inequalities)

Let $X$ be a non-negative random variable that is almost surely bounded but not identically zero, show that

P (X \geq \frac{1}{2} E X) \geq \frac{1}{2} \frac{E X}{∥ X ∥_{\infty}} .