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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 CauchySchwarz Inequality: For
Very brief notes on measures: From σfields to Carathéodory’s Theorem
Definition 1. A field is a nonempty 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
Rio’s Inequality
Let and be two integrable realvalued random variables and let be the quantile function of . Then if is integrable over we have where is the amixing coefficient. Proof: Set and then since and note also that which implies that
Kolmogorov’s Maximal Inequality
Let be independent random variables with . Set . Then Proof: Let Notice that and