Please no written by hand solution and no img 2. (i) Let X ≥ 0 be a ranodm variable such that E(|X|) < ∞. ProveE(X)E(√X)√hP(X ≥ h) ≤≤√ h(ii) If E(2x) = 4 then P(X ≥ 3) ≤ 2.(iii) Let E(X) = µ and Var(X) = ² <∞. Prove that for any a > 0 and y > 0, we haveP(X− µ ≥ a) ≤ P ((X − µ + y)² ≥ (a + y)²)(1).Apply the Markov inequality on the right hand side of the inequality (1) and then minimizeits upper bound in terms of y to conclude:P(X−μ ≥ a) ≤0²0² + a².

Chebyshev and Markov inequalities

2. (i) Let X20 be a ranodm variable such that E(X) <∞. Prove E(√X) √h P(X ≥ h) ≤ E(X) h (ii) If E(2X) = 4 then P(X≥ 3) ≤ (iii) Let E(X) = μ and Var(X) = 0² <∞. Prove that for any a > 0 and y> 0, we have P(X−μ ≥ a) ≤P ((X− μ+ y)² ≥ (a + y)²) (1). Apply the Markov inequality on the right hand side of the inequality (1) and then minimize its upper bound in terms of y to conclude: P(X-μ ≥a) ≤ 0² + a²

2. (i) Let X20 be a ranodm variable such that E(X) <∞. Prove E(√X) √h P(X ≥ h) ≤ E(X) h (ii) If E(2X) = 4 then P(X≥ 3) ≤ (iii) Let E(X) = μ and Var(X) = 0² <∞. Prove that for any a > 0 and y> 0, we have P(X−μ ≥ a) ≤P ((X− μ+ y)² ≥ (a + y)²) (1). Apply the Markov inequality on the right hand side of the inequality (1) and then minimize its upper bound in terms of y to conclude: P(X-μ ≥a) ≤ 0² + a²

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 91E

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