PDF Lecture 3 - Expectation, moments and inequalities For a real-valued random variable X, its expectation E[X] (sometimes just EX) . If X and Y are independent random variables, then If the random variable can take on only a finite number of values, the "conditions" are that . Then, E[XY] = P!2 X(!)Y(!)P(! Calculate moments for joint, conditional, and marginal random variables ... The intuition, in one sentence, is that if you start from . PDF A Conditional expectation - University of Arizona Properties of Expectation. The inequality you have asserted is false: A simple counter-example is X ∼ Bin ( 2, 1 2) and c = 1, which gives you the expectation: E ( max ( X, c)) = 3 4 ⋅ 1 + 1 4 ⋅ 2 = 5 4. Now let μ be the distribution of your random variable X on [e, + ∞). Suppose thatE(X2)<∞andE(Y2)<∞.Hoeffding proved that Cov(X,Y)= R2 Suppose that E(X2)<∞and E(Y2)<∞.Hoeffding proved . . Example 1. In this appendix we present specific properties of the expectation (additional to justtheintegralofmeasurablefunctionsonpossiblyinfinitemeasurespaces).Itis to be expected that on probability spaces we may obtain more specific properties since the probability space has measure 1. Example: Roll a die until we get a 6. STA 711: Probability & Measure Theory Robert L. Wolpert 5 Expectation Inequalities and Lp Spaces Fix a probability space (Ω,F,P) and, for any real number p > 0 (not necessarily an integer) and let \Lp" or \Lp(Ω,F,P)", pronounced \ell pee", denote the vector space of real-valued (or sometimes complex-valued) random variables X for which E|X|p < ∞.
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