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3. Inequalities

3. Inequalities. Markov’s inequality Chebyshev’s inequality Cauchy-Schwartz’s inequality Jensen’s inequality. 1. Markov’s inequality. X>0 ; EX . Then:. Remarks: (i) (ii) small if EX small. 2. INEQUALITIES. Chebyshev’s inequality.

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3. Inequalities

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  1. 3. Inequalities • Markov’s inequality • Chebyshev’s inequality • Cauchy-Schwartz’s inequality • Jensen’s inequality 1

  2. Markov’sinequality • X>0; EX. Then: Remarks: (i) (ii) small if EX small 2 INEQUALITIES

  3. Chebyshev’sinequality X; Then: 3 INEQUALITIES

  4. Cauchy – Schwartz’sinequality X, Y; EX, EY, VX, VY. Then: 4 INEQUALITIES

  5. Jensen’sinequality X random variable; g(•) a convex function. Then: g(•) is convex if g(•) takes values below the line g(x)+(1-)g(y) 5 INEQUALITIES

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