Section8Notes

# Section8Notes - SECTION NOTES YANKUN WANG Variable...

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SECTION NOTES YANKUN WANG Variable Transformation Let X have pdf f X ( x ) and let Y = g ( X ), where g is a monotone function. Let A = x : f X ( x ) > 0, and B = y : y = g ( x ) forsomex A . Suppose f X ( x ) is continuous on A and that g - 1 ( y ) has a continuous derivative on B . Then the pdf of Y is given by: Theorem 1. Let X have pdf f X ( x ) and let Y = g ( X ) , where g is a monotone function. Let mathcalX = { x : f X ( x ) > 0 } and mathcalY = { y : y = g ( x ) for some x ∈ X} . Suppose that f X x is continuous on mathcalX and that g - 1 ( x ) has a continuous derivative on mathcalY . Then the pdf of Y is given by f Y ( y ) = ( f X ( g - 1 ( y )) d dy g - 1 ( y ) y ∈ Y 0 otherwise Example. Let X have the standard normal distribution , f X ( x ) = 1 2 π e - x 2 / 2 , -∞ < x < Consider Y = X 2 . The function g ( x ) = x 2 is monotone on ( -∞ , 0) and on (0 , ). The set Y = (0 , ). We take A 0 = { 0 } ; A 1 = ( -∞ , 0) , g 1 ( x ) = x 2 , g - 1 1 ( y ) = - y ; A 2 = (0 , ) , g 2 ( x ) = x 2 , g - 1 2 ( y ) = y. The pdf of Y is f Y ( y ) = 1 2 π e - ( - y ) 2 / 2 - 1 2 y + 1 2 π e - ( - y ) 2 / 2 1 2 y = 1 2 π 1 y e - y/ 2 , 0 < y < .

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