t*******y
发帖数: 11968 | 1 【 以下文字转载自 Mathematics 讨论区 】
发信人: talkdirty (讲脏话), 信区: Mathematics
标 题: How to prove the bound of Chebyshev's inequality can't be improved.
发信站: BBS 未名空间站 (Thu Oct 9 00:09:00 2008)
How to prove the bound of Chebyshev's inequality can't be improved.
请众牛指点或推荐个连接. 谢谢 | l********s
发帖数: 430 | 2 find an intro statistics book pls | d*********a
发帖数: 255 | 3 for any k > 1, Let P(X=-1)=1/(2k^2) ,P(X=0)=1-1/k^2, P(X=1)=1/(2k^2)
EX=0 VarX=1/k^2 sigma=1/k
Chebyshev's inequality :P(|X-EX|>=k*sigma)<=VarX/(k*sigma)^2
In this case, P(|X-EX|>=k*sigma)=P(|X|>=1)=P(X=-1)+P(X=1)=1/k^2
VarX/(k*sigma)^2=1/k^2
They are equal.
the bound can not be improved.
.
【在 t*******y 的大作中提到】
: 【 以下文字转载自 Mathematics 讨论区 】
: 发信人: talkdirty (讲脏话), 信区: Mathematics
: 标 题: How to prove the bound of Chebyshev's inequality can't be improved.
: 发信站: BBS 未名空间站 (Thu Oct 9 00:09:00 2008)
: How to prove the bound of Chebyshev's inequality can't be improved.
: 请众牛指点或推荐个连接. 谢谢
| t*******y
发帖数: 11968 | 4
thank you very much
【在 d*********a 的大作中提到】
: for any k > 1, Let P(X=-1)=1/(2k^2) ,P(X=0)=1-1/k^2, P(X=1)=1/(2k^2)
: EX=0 VarX=1/k^2 sigma=1/k
: Chebyshev's inequality :P(|X-EX|>=k*sigma)<=VarX/(k*sigma)^2
: In this case, P(|X-EX|>=k*sigma)=P(|X|>=1)=P(X=-1)+P(X=1)=1/k^2
: VarX/(k*sigma)^2=1/k^2
: They are equal.
: the bound can not be improved.
:
: .
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