Markov and Chebyshev Inequalities and the Weak Law of Large Numbers

https://www.math.wustl.edu/~russw/f10.math493/chebyshev.pdf

http://www.tkiryl.com/Probability/Chapter%208.pdf

http://mathworld.wolfram.com/ChebyshevInequality.html

Apply Markov's inequality with a=k^2 to obtain

P[(x-mu)^2>=k^2]<=(<(x-mu)^2>)/(k^2)=(sigma^2)/(k^2).
(1)

Therefore, if a random variable x has a finite mean mu and finite variance sigma^2, then for all k>0,

P(|x-mu|>=k) <= (sigma^2)/(k^2)
(2)
P(|x-mu|>=ksigma) <= 1/(k^2).
(3)
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原文地址：https://www.cnblogs.com/rsapaper/p/7808148.html