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Statistics 510: Notes 26 Reading: Section 8.5 (The One Sided Chebyshev Inequality) Chebyshev’s Inequality: If X is a random variable with 2 finite mean and variance , then for any value a 0 , P{| X | a} 2 a2 . Chebyshev’s Inequality provides a bound for a two-sided probability – what if we are only interested in a one-sided probability, i.e., P( X a) or P( X a) ? We can obtain a sharper bound. One-sided Chebyshev’s Inequality: If X is a random variable with finite mean and variance 2 , then for any value a 0 , P{ X a} 2 2 a2 2 . P{ X a} 2 a2 Proof: We first consider the case 0 , i.e., X is a random variable with mean 0. Let b 0 and note that X a is equivalent to X b a b . Hence, P ( X a ) P ( X b a b) P{( X b)2 (a b)2 } 1 where the inequality is obtained by noting since a b 0 , 2 2 we have that X b a b implies ( X b) (a b) . Upon applying Markov’s inequality, the preceding yields that E[( X b)2 ] 2 b2 P( X a ) 2 (a b) (a b)2 . 2 b2 2 The value of b that minimizes (a b)2 is b / a , yielding for a 0 P( X a) 2 2 a2 for X with mean 0 (1.1) Now suppose X has mean , which might not equal 0, 2 and variance Then X and X have mean 0 with 2 variance and we can apply (1.1) to obtain P( X a) 2 2 a2 2 P( X a) 2 a2 Rearranging these inequalities gives P{ X a} 2 2 a2 2 . P{ X a} 2 a2 Example 1: The mean of a list of a million numbers is 10 2 and the mean of the squares of the numbers is 101. Find an upper bound on how many of the entries in the list are 14 or more. In Notes 24, we used the two-sided Chebyshev’s Inequality to find a bound of 62,500. How much better can we do with the one-sided Chebyshev’s Inequality? 3 Follow-up courses to Stat 510: Statistics courses Stat 511: Mathematical statistics (e.g., efficient point estimation, Neyman-Pearson theory of optimal hypothesis testing and statistical methodology (e.g., regression and ANOVA). Spring, MW 10:30-12. Stat 512: Mathematical statistics. Compared to 511, 512 goes more into depth into mathematical statistics and has less focus on statistical methodology. Spring, TTh 3:004:30. Probability courses: Stat 530: Probability theory. Rigorous mathematical course that would prove in generality and rigor the results we proved here without focusing on applications, e.g., measure theory would be discussed which can deal with continuous random variables whose distribution is not smooth enough to have a density and the central limit theorem would be proved without assuming that a random variable has a moment generating function using characteristic functions. Offered in the fall. OPIM 930: Stochastic processes. Good follow-up to this course for students wanting to learn more probability models (e.g., Markov chains) at the level of mathematical rigor of this course. Offered in the fall. 4