Tag Archives: Probability

Expectation: Useful properties and inequalities

If is a random variable on . The expected value of is defined as Inequalities Jensen’s inequality. If is convex and Holder’s inequality. If with then Cauchy-Schwarz Inequality: For

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A nice chart of univariate distribution relationships

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Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A -field is a non-empty collection of subsets of the sample space closed under the formation of complements and countable unions (or equivalently of countable intesections – note ). Hence is a -field if whenever whenever Definition 2. … Continue reading

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Rio’s Inequality

Let and be two integrable real-valued random variables and let be the quantile function of . Then if is integrable over we have where is the a-mixing coefficient. Proof: Set and then since and note also that which implies that

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Kolmogorov’s Maximal Inequality

Let be independent random variables with . Set . Then Proof: Let Notice that and

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