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st371-23 fns of rv 03

st371-23 fns of rv 03 - ST371 Introduction to Probability...

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(c) Tom Gerig ST371-xx Page 1 ST371 Introduction to Probability and Distribution Theory Distributions of Functions of Random Variables Method of Transformations

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(c) Tom Gerig ST371-xx Page 2 Method of Transformations Find the of the ( ). pdf rv U h Y Let be a with ( ). Let ( ) be a (strictly) increasing or decreasing function of y, over the range of . Y Y rv pdf f y h y y 1 2 1 2 A function ( ) is strictly increasing if: implies ( ) ( ) h y y y h y h y
(c) Tom Gerig ST371-xx Page 3 Method of Transformations Let be a with known ( ) Y Y rv pdf f y Let ( ) be a strickly increasing (SI) or decreasing (SD) function of y, over the range of . h y y Find the of the ( ). pdf rv U h Y

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(c) Tom Gerig ST371-xx Page 4 Method of Transformations ( ) { } { ( ) } U F u P U u P h Y u 1 1 { ( )} ( ( )) SI Y P Y h u F h u Using the method of distribution functions: Let ( ), where ( ) is . U h Y h y SI 1 1 where ( ) is the inverse of function ( ). . . ( ) ( ) h u h y i e u h y y h u
(c) Tom Gerig ST371-xx Page 5 Method of Transformations 1 ( ) ( ( )) U Y F u F h u 1 ( ) ( ( )) ( ) U Y U dF u dF h u f u du du 1 1 1 ( ) ( ( )) ( ( )) Y Y dh u dy f h u f h u du du For ( ) : h y SI 1 Thus for ( ) , ( ) ( ( )) U Y dy f h y SI u f h u du

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(c) Tom Gerig ST371-xx Page 6 Method of Transformations Thus for a function ( ), the of SI h y pdf ( ) U h Y is 1 ( ) ( ( ) ) U Y dy f u f h u du
(c) Tom Gerig ST371-xx Page 7 Method of Transformations 1 ( ) [1 ( ( ))] ( ) U Y U dF u d F h u f u du du ( ) { } { ( ) } U F u P U u P h Y u 1 1 { ( )} 1 ( ( )) SD Y P Y h u F h u

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st371-23 fns of rv 03 - ST371 Introduction to Probability...

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