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HelpfulTheorems-4

HelpfulTheorems-4 - Markovs Inequality Inequality...

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1 Inequality, Probability, and Joviality In many cases, we don’t know the true form of a probability distribution E.g., Midterm scores But, we know the mean May also have other measures/properties o Variance o Non-negativity o Etc. Inequalities and bounds still allow us to say something about the probability distribution in such cases o May be imprecise compared to knowing true distribution! Markov’s Inequality Say X is a non-negative random variable Proof: I = 1 if X ≥ a , 0 otherwise Taking expectations: 0 , ] [ ) ( a a X E a X P all for , 0 a X I X Since a X E a X E a X P I E ] [ ) ( ] [ Andrey Andreyevich Markov Andrey Andreyevich Markov (1856-1922) was a Russian mathematician Markov’s Inequality is named after him He also invented Markov Chains… o …which are the basis for Google’s PageRank algorithm His facial hair inspires fear in Charlie Sheen Markov and the Midterm Statistics from last quarter’s CS109 midterm X = midterm score Using sample mean X = 80.9 E[X] What is P(X ≥ 91)? Markov bound: 88.92% of class scored 91 or greater In fact, 36.67% of class scored 91 or greater o Markov inequality can be a very loose bound o But, it made no assumption at all about form of distribution! 8892 . 0 91 9 . 80 91 ] [ ) 91 ( X E X P
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