Show that for any random variable $X$, and any $a > 0$, $$P(|X| > a) \leq {EX^4 \over a^4}.$$

Maybe I need to use Markov's Inequality, but I don't know how.

  • 4
    $\begingroup$ Hint: $|X| > a$ iff $|X|^4 > a^4$. $\endgroup$ – Nate Eldredge Jul 27 '15 at 4:53

Hint: $$1_{\{|X|>a\}} \leq \left| \frac{X}{a} \right|^4 1_{\{|X|>a\}} \leq \left| \frac{X}{a} \right|^4. $$ Now integrate both sides.


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