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I have found some examples on the web for specific random variables such $X$ (a discrete type) with probabilities $1/8$, $3/4$ and $1/8$ at the points $x=-1,0,1$ with $\mu=0$ and $\sigma=1/2$. Then, when $k=2, 1/k^2=1/4$ and $\Pr(|X-\mu|\ge k\sigma)=\Pr(X\ge1)=1/4$, so it attains the upper bound $1/4$ and it can´t be improved without further assumptions. I would like to know if there is a way or if you have any ideas to prove that the inequality cannot be improved without further assumptions about the distribution of any random variable

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Try a discrete distribution with probabilities $\dfrac1{2k^2},1-\dfrac1{k^2},\dfrac1{2k^2}$ at points $-k\sigma,0,k\sigma$.

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