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Category Archives: Probability
Karl Popper: Conjectures and Refutations
(1) It is easy to obtain confirmations, or verifications, for nearly every theoryif we look for confirmations. (2) Confirmations should count only if they are the result of risky predictions; that is to say, if, unenlightened by the theory in … Continue reading
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