Chapter 10. Transformation of Random Variables

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

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Unformatted text preview: § 10 Transformation of random variables § 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 ) ≤ y ) . 10.1.2 For discrete X 1 ,...,X n with joint mass function f ( x 1 ,...,x n ), F Y ( y ) = ∑ ··· ∑ { ( x 1 ,...,x n ): g ( x 1 ,...,x n ) ≤ y } f ( x 1 ,...,x n ) , and the corresponding mass function of Y is f Y ( y ) = ∑ ··· ∑ { ( 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 ··· R { ( x 1 ,...,x n ): g ( x 1 ,...,x n ) ≤ y } f ( x 1 ,...,x n ) dx 1 ··· dx n , and the corresponding pdf of Y is f Y ( y ) = F Y ( y ) . § 10.2 Transformation of one random variable: examples 10.2.1 X : random variable with cdf F X and pdf f X , α,β : constants with α 6 = 0. Let Y = αX + β . F Y ( y ) = P ( αX + β ≤ y ) = P X ≤ y- β α ¶ = F X y- β α ¶ , α > , P X ≥ y- β α ¶ = 1- F X y- β α ¶ , α < , f Y ( y ) = 1 | α | f X y- β α ¶ . 70 Special case : If X ∼ N ( μ,σ 2 ), then f Y ( y ) = 1 √ 2 πσ 2 α 2 exp ‰- ( y- ( αμ + β )) 2 2 α 2 σ 2 , so that Y ∼ N ( αμ + β, α 2 σ 2 ). [ Recall: E [ Y ] = α E [ X ] + β = αμ + β and Var( Y ) = α 2 Var( X ) = α 2 σ 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 ≤ y ) = P ( X ≥ 1 /y ) = 1- F X (1 /y ) , y ≥ , , y < ....
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Chapter 10. Transformation of Random Variables - § 10...

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