Lecture 04-mathematical expectation

# Lecture 04-mathematical expectation - 1036 Probability...

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Prob. & Stat. Lecture04 - mathematical expectation 4-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 4 Lecture 4 Mathematical Mathematical Expectation Expectation

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Prob. & Stat. Lecture04 - mathematical expectation 4-2 Mean of a Random Variable •L e t X be a random variable with probability distribution f ( x ). The mean or expected value of X is •E x a m p l e : – If 2 coins are tossed 16 times. The outcomes are 0 head: 4 times; 1 head: 7 times; 2 heads: 5 times. The average number of heads per toss? ( ) = = x x xf X E ) ( µ () = = dx x xf X E ) ( if X is discrete, and if X is continuous 06 . 1 16 5 ) 2 ( 16 7 ) 1 ( 16 4 ) 0 ( 16 ) 5 )( 2 ( ) 7 )( 1 ( ) 4 )( 0 ( = + + = + +
Prob. & Stat. Lecture04 - mathematical expectation 4-3 Example 4.3 •L e t X be the random variables that denotes the life in hours of a certain electronic device. The probability density function is Find the expected life of this type of device. Solution > = eleswhere , 0 100 , 000 , 20 ) ( 3 x x x f () 200 20000 100 3 = = = dx x x X E µ

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Prob. & Stat. Lecture04 - mathematical expectation 4-4 Expectation of g ( x ) •L e t X be a random variable with probability distribution f ( x ). The mean or expected value of the random variable g ( x ) is = = = = dx x f x g X g E x f x g X g E X g X g ) ( ) ( )] ( [ ) ( ) ( )] ( [ ) ( ) ( µ If X is discrete If X is continuous Ex: X is a RV with pdf: elsewhere 2 x 1 - , 0 , 3 ) ( 2 < < = x x f The mean of g ( X )=4 X +3 ??? [] () 8 3 3 4 ) ( ) ( 3 4 2 1 2 = + = = + dx x x dx x f x g X E
Prob. & Stat. Lecture04 - mathematical expectation 4-5 Expectation of g ( X,Y ) •L e t X and Y be two random variables with joint probability distribution f ( x, y ). The mean or expected value of the random variable g ( X,Y ) is ∫∫ ∑∑ == = = = dxdy y x f y x g Y X g E y x f y x g Y X g E Y X g xy Y X g ) , ( ) , ( )] , ( [ ) , ( ) , ( )] , ( [ ) , ( ) , ( µ If X,Y are discrete If X,Y are continuous Ex: Find E( Y/X ) for the density function ( ) elsewhere 1 0 , 2 0 , 0 , 4 3 1 ) , ( 2 < < < < + = y x y x y x f ( ) 8 5 4 3 1 2 1 0 2 0 = + = dxdy y x x y X Y E Sol:

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Prob. & Stat. Lecture04 - mathematical expectation 4-6 Example •L e t X and Y be random variables with joint density function elsewhere. 1 y 1,0 x 0 , 0 , 4 ) , ( < < < < = xy y x f Find the expected value of 2 2 Y X Z + = () ∫∫ = + = = 1 0 1 0 2 2 4 ) , ( xydxdy y x dxdy y x zf Z E Solution:
Prob. & Stat. Lecture04 - mathematical expectation 4-7 Remark () ∫∫ = ∑∑ = = = dx x xg dxdy y x xf x xg y x f x y x xf X E xx y x y ) ( ) , ( ) ( ) , ( ) , ( discrete continuous = = = = dy y yh dxdy y x xf y yh y x f y y x yf Y E xy x y y ) ( ) , ( ) ( ) , ( ) , ( E ( X ) & E ( Y ) calculated by joint pdf or marginal pdf discrete continuous

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Prob. & Stat. Lecture04 - mathematical expectation
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Lecture 04-mathematical expectation - 1036 Probability...

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