What is Chebyshev's inequality


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Chebyshev inequality; Markov's inequality

Theorem 4.18 (Chebyshev inequality) Be a random variable with
Then applies to each
(72)

proof
 
  • From the linearity or monotonicity properties of the expected value (see Theorem 4.4) it follows that for each


     
     


Notice
 
Examples
 
  1. incorrect measurements 
    • It is known from a measuring device that the measurement results are error-prone.
    • The -th measurement of an (unknown) quantity deliver the value For .
    • The measurement errors be independent and identically distributed random variables.
    • About the distribution of just know that
      and (74)

    • The question of how many measurements are required to be able to conclude with the help of the Chebyshev inequality (72) that the arithmetic mean
      of the random readings at most with probability more than of the "true" but unknown value deviates.
    • From the elementary properties of the expected value and the variance of the sum of independent random variables results (cf. Corollary 4.2 and theorems 4.6 and 4.10) that

      and

    • From this and from the Chebyshev inequality (72) it follows that
    • So it applies if .
    • From these considerations it follows that the above-mentioned accuracy requirements are met, if Measurements are made.
  2. normally distributed measurement errors
Notice
 


Next:Types of convergence and limit theorems Up:Inequalities for moments and Previous:Jensen Inequality & nbsp Contents Ursa Pantle 2004-05-10