Important Inequalities - Chebyshev Inequality, Cauchy-Schwarz Inequality <proof, applications, covariance Inequality>

Chebyshev Inequality

 

Chebyshev Inequality is one of the most important inequalities highly based on Markov's Inequality.

It is easier to memorize 2nd notation to apply.

 

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proof 

you can easily have Chebyshev Inequality by using applications of Markov's Inequality so if you want to perfectly understand the Inequality I recognize you to study Markov's Inequality 

[Statistics] - Markov's Inequality 마르코프 부등식

 

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Examples

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Cauchy-Schwarz Inequality 

 three Inequalities below look different but they are under Cauchy-Schwarz Inequality. 

First notation is usually used in Linear Algebra, Second and Third notation is usually used in Statistics. 

So I will apply second and third notation for this post. 

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proof

1. 

proof

2. 

proof 

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equality condition : X and Y are linearly dependent

two sides of the Ineqaulity are equal if and only if X and Y are linearly dependent Y=tX

                                                             if and only if a and b are linearly dependent b=ta

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Application - covariance Inequality