08_24 - STAT 410 Examples for 08/24/2011 expected value E(...

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STAT 410 Examples for 08/24/2011 Fall 2011 expected value E ( X ) = μ X discrete continuous If x x p x all ) ( < , E ( X ) = x x p x all ) ( If - x x f x d ) ( < , E ( X ) = - x x f x d ) ( Example 1: x p ( x ) x p ( x ) E ( X ) = μ X = 2.3. 1 0.2 0.2 2 0.4 0.8 3 0.3 0.9 4 0.1 0.4 2.3 Example 2: f X ( x ) = < < o.w. 0 1 0 3 2 x x E ( X ) = μ X = 1 0 2 3 x x x d = 1 0 3 3 x x d = 4 3 = 0.75. Example 3: f X ( x ) = > - o.w. 0 1 5 6 x x E ( X ) = μ X = - 1 6 5 x x x d = - 1 5 5 x x d = 4 5 = 1.25.
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Example 4: ( Standard ) Cauchy distribution: f X ( x ) = ( ) 2 1 1 x + π , < x < . Even though f X ( x ) is symmetric about zero, E ( X ) is undefined since ( ) + - dx x x 1 1 2 = . ----------------------------------------------------------------------------------------------------------------- discrete continuous If x x p x g all ) ( ) ( < , E ( g ( X ) ) = x x p x g all ) ( ) ( If - x x f x g d ) ( ) ( < , E ( g ( X ) ) = - x x f x g d ) ( ) ( variance Var ( X ) = 2 X σ = E ( [ X - μ X ] 2 ) = E ( X 2 ) [ E ( X ) ] 2 discrete continuous Var ( X ) = ( ) - x x p x all 2 X ) ( μ = [ ] 2 all 2 ) X ( E ) ( x x p x - Var ( X ) = ( ) - - x x x d f ) ( 2 X μ = [ ] 2 2 ) X ( E ) ( x x x d f - - Example 1: x p ( x ) x 2 p ( x ) E ( X 2 ) = 6.1 Var ( X ) = 6.1 – 2.3 2 = 0.81 1 0.2 0.2 2 0.4 1.6 3 0.3 2.7 4 0.1 1.6 6.1
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Example 2: f X ( x ) = < < o.w. 0 1 0 3 2 x x E ( X 2 ) = 1 0 2 2 3 x x x d = 1 0 4 3 x x d = 5 3 . Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 2 4 3 5 3 - = 80 3 . Example 3: f X ( x ) = > - o.w. 0 1 5 6 x x E ( X 2 ) = - 1 6 2 5 x x x d = - 1 4 5 x x d = 3 5 . Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 2 4 5 3 5 - = 48 5 . ----------------------------------------------------------------------------------------------------------------- The k th moment of X ( the k th moment of X about the origin ), μ k , is given by μ k = E ( X k ) = ( ) x k x p x all OR ( ) - dx x f x k The k th central moment of X ( the k th moment of X about the mean ), μ k ' , is given by μ k ' = E ( ( X – μ ) k ) = ( ) ( ) - x k x p x all μ OR ( ) ( ) - - μ dx x f x k
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moment-generating function M X ( t ) = E ( e t X ) discrete continuous M X ( t ) = ( ) x x t x p e all M X ( t ) = ( ) - dx x f x t e Theorem 1 : M X 1 ( t ) = M X 2 ( t ) for some interval containing 0 f X 1 ( x ) = f X 2 ( x ) Theorem 2 : M X ' ( 0 ) = E ( X ) M X " ( 0 ) = E ( X 2 ) M X ( k ) ( 0 ) = E ( X k ) Theorem 3 : Let V = a X + b . Then M V ( t ) = e b t M X ( a t ) Example 1: x p ( x ) M X ( t ) = E ( e t X ) = 0.2 e t + 0.4 e 2 t + 0.3 e
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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08_24 - STAT 410 Examples for 08/24/2011 expected value E(...

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