$X$ is a random variable of any sign, such that $\mathbb{E}(|X|^k) $ exists for $k$ positive integer.
My question is: how to show that Markov's Markov inequality can be generalized to the following form
$$ P(|X>\epsilon |) \leq \frac{\mathbb{E}(|X|^k)}{\epsilon^k} $$
Best Answer
Hint:-
For positive numbers, is $a>b$ equivalent to $a^{p}>b^{p}$ ? If yes then what can you say about the event $\{|X|>c\}$ and $\{|X|^{p}>c^{p}\}$ ? . What can you then say about the probabilities of the above events? What can you then do with Markov's inequality ?