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)

 

 

posted @ 2017-11-09 10:30  papering  阅读(217)  评论(0)  编辑  收藏  举报