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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 (IE@CUHK) 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 (IE@CUHK) 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...
View Full Document

Ask a homework question - tutors are online