Chapter 10. Transformation of Random Variables

# Chapter 10. Transformation of Random Variables - Â 10...

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Unformatted text preview: Â§ 10 Transformation of random variables Â§ 10.1 General principle 10.1.1 Let X 1 ,...,X n be n random variables. Define Y = g ( X 1 ,...,X n ) for some real-valued function g . Then the distribution of Y can be described by its cdf F Y ( y ) = P ( g ( X 1 ,...,X n ) â‰¤ y ) . 10.1.2 For discrete X 1 ,...,X n with joint mass function f ( x 1 ,...,x n ), F Y ( y ) = âˆ‘ Â·Â·Â· âˆ‘ { ( x 1 ,...,x n ): g ( x 1 ,...,x n ) â‰¤ y } f ( x 1 ,...,x n ) , and the corresponding mass function of Y is f Y ( y ) = âˆ‘ Â·Â·Â· âˆ‘ { ( x 1 ,...,x n ): g ( x 1 ,...,x n )= y } f ( x 1 ,...,x n ) . 10.1.3 For continuous X 1 ,...,X n with joint pdf f ( x 1 ,...,x n ), F Y ( y ) = R Â·Â·Â· R { ( x 1 ,...,x n ): g ( x 1 ,...,x n ) â‰¤ y } f ( x 1 ,...,x n ) dx 1 Â·Â·Â· dx n , and the corresponding pdf of Y is f Y ( y ) = F Y ( y ) . Â§ 10.2 Transformation of one random variable: examples 10.2.1 X : random variable with cdf F X and pdf f X , Î±,Î² : constants with Î± 6 = 0. Let Y = Î±X + Î² . F Y ( y ) = P ( Î±X + Î² â‰¤ y ) = ï£± ï£´ ï£´ ï£´ ï£² ï£´ ï£´ ï£´ ï£³ P X â‰¤ y- Î² Î± Â¶ = F X y- Î² Î± Â¶ , Î± > , P X â‰¥ y- Î² Î± Â¶ = 1- F X y- Î² Î± Â¶ , Î± < , f Y ( y ) = 1 | Î± | f X y- Î² Î± Â¶ . 70 Special case : If X âˆ¼ N ( Î¼,Ïƒ 2 ), then f Y ( y ) = 1 âˆš 2 Ï€Ïƒ 2 Î± 2 exp â€°- ( y- ( Î±Î¼ + Î² )) 2 2 Î± 2 Ïƒ 2 , so that Y âˆ¼ N ( Î±Î¼ + Î², Î± 2 Ïƒ 2 ). [ Recall: E [ Y ] = Î± E [ X ] + Î² = Î±Î¼ + Î² and Var( Y ) = Î± 2 Var( X ) = Î± 2 Ïƒ 2 , which conform with the above result. ] 10.2.2 X : positive random variable with cdf F X and pdf f X . Let Y = 1 /X . F Y ( y ) = ( P (1 /X â‰¤ y ) = P ( X â‰¥ 1 /y ) = 1- F X (1 /y ) , y â‰¥ , , y < ....
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## This note was uploaded on 04/29/2010 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Chapter 10. Transformation of Random Variables - Â 10...

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