# V uniformly distributed in 0 4 e x 2 and varx 43

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Unformatted text preview: G2430C lecture 10 Var(Y ) 1 =. 4 4 9 / 29 Chebyshev Inequality Chebyshev bound is tighter than Markov bound. [Example 7.2 in textbook] Let X be a r.v. uniformly distributed in [0, 4] E [X ] = 2 and Var(X ) = 4/3 Applying Chebyshev inequality, we have 1 P (X ≥ 2) = P (|X − 2| ≥ 0) ≤ 2 1 P (X ≥ 3) = P (|X − 2| ≥ 1) ≤ 2 1 P (X ≥ 4) = P (|X − 2| ≥ 2) ≤ 2 1 (applying the idea, tight) 2 1 Var(X ) 2 = = 0.67 (loose) 21 3 1 Var(X ) 1 = (loose) 24 6 Chebyshev inequality utilizes both mean and variance. M. Chen ([email protected]) ENGG2430C lecture 10 10 / 29 Markov Inequality vs Chebyshev Inequality Markov inequality nonnegative r.v.s Chebyshev inequality r.v.s utilizes mean utilizes both mean and variance P (X ≥ a) ≤ E [X ] a loose bound P (|X − µ | ≥ c ) ≤ σ2 c2 tight bound Even tighter bound? Chernoﬀ inequality! Utilize all orders of statistics (moments). M. Chen ([email protected]) ENGG2430C lecture 10 11 / 29 ATV poll problem: Observation Deﬁne Xi as a Bernoulli r.v. indicating the answer of i -th family pXi (1) = p , pXi (0) = 1 − p , E [Xi ] = p...
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