Section3.3-students_SP11

Section3.3-students_SP11 - Recap (Probability) Distribution...

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1 Today …. • Expected Value (“mean”) of h(X) : • Variance of X : Recap … (Probability) Distribution of X means the specification of: P robability M ass F unction (pmf of X) : C umulative D istribution F unction (cdf of X): p X (x) = P( X = x) = P ( { s ε S : X(s) = x } ) F X (x) = P( X ≤ x) = P ( { s ε : X(s) ≤ x } ) Some numerical summary measures of the distribution of X: Section 3.3 Expected Value of Discrete RV X Uses: 1. Provides a numerical summary of a distribution 2. Typical value of X if the experiment is repeated many times Example: Pay $50 to play the following game: Spin, X = outcome Section 3.3
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2 Question: If we repeatedly play the game, what is our long term average net profit? Specifically, say n = 1,000,000 = 10 6 times. Average = Section 3.3 Expected Values Meaning: X 1 , X 2 , ... are independent “draws” from p X (.) with corresponding h ( X 1 ), h ( X 2 ) , . .. Then E [ h(X) ] is the long term average of the { h ( X i )} Def: X is a discrete RV with pmf p X (.) and h ( . ) is a real-valued function. The expected value of h ( X ) is Section 3.3
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3 Mean as “center of mass” E[X] is one measure of the “center” of a probability distribution.
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This document was uploaded on 07/26/2011.

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Section3.3-students_SP11 - Recap (Probability) Distribution...

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