Markov inequality when r-th momentum of $x$ exists.

50 Views Asked by At

Thank you for taking the time to read my post. I greatly value any advice you might be able to provide.

I've recently learned about Markov inequalities in my probability and statistics course.

$P\left(\left|X\right| \geq k\right) \leq \frac{\mathbb{E}\left(\left|X\right|^r\right)}{k^r}$

This is my question

Assume that the rth moment of x exists for every positive real number r. In this case, which value of r should I select to establish an upper bound?

I apologize if my English is not perfect; I'm still learning. If there are any parts of my questions that are unclear, please feel free to let me know. I appreciate your understanding and patience