06derived - → 1-1) and differentiable function. 2a. Pr [...

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EECS 501 DERIVED DISTRIBUTIONS Fall 2001 Given: pdf f x ( X ) and function y = g ( x ). Goal: Compute pdf f y ( Y ). pmfs: p y ( Y ) = Pr [ y = Y ] = Pr [ g ( x ) = Y ] = Pr [ x g - 1 ( Y )] = g - 1 ( Y ) p x ( X ). Scale: y = ax f y ( Y ) = 1 | a | f x ( Y a ) so integrates to one. Compare to δ ( X ). Shift: y = x - b f y ( Y ) = f x ( Y + b ) just shift pdf. I. Method of events: Straightforward; g ( · ) need not be differentiable. 1. F y ( Y ) = Pr [ y Y ] = Pr [ g ( x ) Y ] = Pr [ x g - 1 ( { y : y Y } )]. 2. f y ( Y ) = d dY F y ( Y ) = d dY R X g - 1 ( { y : y Y } ) f x ( X ) dX . EX1: f x ( X ) = 1 2 , 0 < X < 2; 0 otherwise. y = g ( x ) = 1 /x . Compute f y ( Y ). 1. F y ( Y ) = Pr [ y Y ] = Pr [ 1 x Y ] = Pr [ x 1 Y ]. X < 2 Y > 1 2 . 2. F y ( Y ) = 1 2 (2 - 1 Y ) if Y > 1 2 0 if Y < 1 2 f y ( Y ) = 1 / (2 Y 2 ) if Y > 1 2 0 if Y < 1 2 . 3. Check: F y ( Y ) is continuous at Y = 1 2 ; F y ( -∞ ) = 0; F y ( ) = 1. EX2: Arbitrary f x ( X ). y = | x | . Compute f y ( Y ) in terms of f x ( X ). 1. F y ( Y ) = Pr [ y Y ] = Pr [ | x | ≤ Y ] = Pr [ - Y x Y ] = R Y - Y f x ( X ) dX . 2. f y ( Y ) = d dY R Y - Y f x ( X ) dX = f x ( Y ) + f x ( - Y ) ,Y 0; 0 if Y < 0. II. Jacobian method: Requires g ( · ) to be differentiable, but easier. 1. Suppose g ( · ) is any nondecreasing (
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Unformatted text preview: → 1-1) and differentiable function. 2a. Pr [ a < x < b ] = Pr [ g ( a ) < y < g ( b )] = R g ( b ) g ( a ) f y ( Y ) dY . 2b. Pr [ a < x < b ] = R b a f x ( X ) dX = R g ( b ) g ( a ) f x ( X = g-1 ( Y )) | dx/dy | dY . 2c. These are equal if f y ( Y ) = f x ( X = g-1 ( Y )) | dg-1 ( Y ) /dY | . EX1: y = 1 /x → X = g-1 ( Y ) = 1 /Y is decreasing and 1-1 for X,Y > 0. f y ( Y ) = | d (1 /Y ) dY | ‰ 1 2 if Y > 1 2 otherwise = ‰ 1 / (2 Y 2 ) if Y > 1 2 otherwise . EX2: Can’t use Jacobian since y = | x | not differentiable at x = 0....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

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06derived - → 1-1) and differentiable function. 2a. Pr [...

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