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# slide4 - ECON321 Econometrics Lecture 4 Continuous Random...

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ECON321 : Econometrics Lecture 4 : Continuous Random Variables and Probability Distributions Sasan Bakhtiari University of Maryland, College Park Summer 2007 Session II,

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Continuous Probability Distribution I The probability of a single value happening is zero. I Instead, f ( x ) is the probability density function (pdf). I P ( a X b ) = R b a f ( x ) dx . I The cumulative distribution function (cdf) F ( x ) = P ( X x ) = Z x -∞ p ( y ) dy . Properties of f ( x ) I f ( x ) 0 , x R , I R -∞ f ( x ) dx = 1. I F ( -∞ ) = 0 , F ( ) = 1.
Example 4.1 Assume that f ( x ) = ax 2 , x [ - 1 , 1] 0 , otherwise For f ( x ) to be a density function we must have Z -∞ f ( x ) = 1 Z 1 - 1 ax 2 dx = a x 3 3 1 - 1 = 2 a 3 = 1 a = 3 / 2 I What is the probability that X 1 / 2? P ( X 0) = Z 1 1 / 2 3 2 x 2 dx = x 3 2 1 1 / 2 = 7 16 . I What is the median? Z m - 1 3 2 x 2 dx = 1 2 m = 0 .

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Expected Value and Variance Expected Value: E ( X ) = R -∞ xf ( x ) dx . General Expected Value: E ( h ( X )) = R -∞ h ( x ) f ( x ) dx , Variance: V ( X ) = σ 2 X = E ( X - E ( X )) 2 = E ( X 2 ) - [ E ( X )] 2 . also I If h ( X ) = aX + b (a linear form) then E ( aX + b ) = aE ( x ) + b . I V ( aX + b ) = a 2 V ( X ).
Example 4.2 For the density function f ( x ) = 3 2 x 2 , x [ - 1 , 1] : I E ( X ) = R 1 - 1 x .

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