notes_for_28th_feb - 2 2(X p)2 and let r = k ox for a given...

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Chebyshev's Theorem (or Chebyshev's Inequality) shows an interesting relationship between p and 0. First Marlcov's Theorem answers the question: If X is a non-negative random variable (i.e. Sx = {x I x 2 0) ), what is the largest possible value of P(X 2 r) for a given r > O? The theorem's answer to the question is P(X 2 r) I y / r where y = E(X). Chebvshev's Theorem says let the non-negative random variable
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Unformatted text preview: 2 2 (X - p)2 and let r = k ox for a given k > 0. Then Markov's theorem says Usually, the theorem is written in the equivalent forrn 1 P(IX-pl2ko)G--- k2 Proof of Markov's Theorem: By definition p = C xf (x) = C xf (x) + C xf (x) 2 C xf (x) , and all {xlx<r> {xlx>r) {xIx'r) X...
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