Show that Markov's inequality is as tight as it possible. Given a positive integer $k$, describe a random variable $X$ that assumes only non-negative values:

$$\Pr[X \geq k E[X] ] = 1/k.$$

Using Markov's bound, we can show at most $1/k$. But how to show equality?! My question to be exact what is the idea to prove the tightness of this bound!


Consider a random variable $X$ which takes the value $1$ with probability $1/k$ and $0$ with probability $1-1/k$.

  • $\begingroup$ Thank you for your answer. It really helps me. I understand that "you give example" to show that it is tight so there is "no proof" as I understand. Also, I understand that when expected value is 1, then we have a tight bound for Markov's bound. So, thank you, I was thinking that there might be a proof. $\endgroup$ – YOUSEFY Sep 19 '17 at 6:21
  • $\begingroup$ To @Davide : Maybe you meant "$X$ which takes the value $1$ with probability $1/k$"? $\endgroup$ – Arundo Donax Sep 28 '17 at 17:00
  • $\begingroup$ @ArundoDonax You are right. $\endgroup$ – Davide Giraudo Sep 28 '17 at 19:00
  • $\begingroup$ @DavideGiraudo I believe X should take the value "k" not "1", with probability 1/k. $\endgroup$ – Toastgeraet Feb 17 at 14:20

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