I guess this is not a difficult problem, but I can not find the function for which to use Chebyshev's inequality...

Assume X is a random variable with property that . Use Chebyshev's inequality for the random variable to prove that

for .

Thanks.

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- Jun 7th 2010, 01:56 PMVeveChebyshev's inequality
I guess this is not a difficult problem, but I can not find the function for which to use Chebyshev's inequality...

Assume X is a random variable with property that . Use Chebyshev's inequality for the random variable to prove that

for .

Thanks. - Jun 7th 2010, 02:07 PMLaurent
You have, using Chebyshev's inequality, for , . Now, expand the square inside the expectation; then the right-hand side is a function of , while the left-hand side isn't. The "best" (sharpest) inequality is the one when the right-hand side is lowest, so you have to choose which minimizes the right-hand side. This resumes to a standard function study (derivative, etc.) to find this minimum.