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Let us say that we have a random probability distribution and we know it's Mean and it's Standard Deviation.

My question is that, is it necessarily true that the majority of values (>= 50%) in that particular probability distribution will lie between: -

$$\mu \pm 1 \sigma$$

Are there any probability distribution where this is not the case? Also is there any proof for the same?

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  • $\begingroup$ Try $P(X=-k)=P(X=+k)=0.4999$ and $P(X=0)=0.0002$ and find $\mu\pm 1\sigma$ $\endgroup$
    – Henry
    Sep 29, 2022 at 16:13
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    $\begingroup$ This may be relevant to your interests: Chebyshev's inequality. $\endgroup$
    – Brian Tung
    Sep 29, 2022 at 16:17

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