Function Of One RV - FUNCTIONS OF ONE RANDOM VARIABLE...

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FUNCTIONS OF ONE RANDOM VARIABLE Problem Statement Assume we know the pdf ( ) of a random variable . Consider a new random variable ( ). We want to find ( ). X Y fx X Y gX fy = Method: Find the cdf of Y first and then the pdf for continuous random variables.
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Linear Function Let Y aX b = . Assume is a continuous rv and 0 Xa Solution For 0, () [ ] [] () Y X a F y PY y yb PX a F a > = = = For 0, ] [ ] 1( ) Y X a a F a < = ≤= = X Y 0 Y aX b a = > y a
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For 0, ( ) ( ) () ( )( ) ()() ( ) 1 ( ) ( )0 0 1 YX YY X yb a X a XX X a X a Fy F a f y y F ya f x dx y f f f x dx a y y f f dx aa f −∞ −∞ −∞ >= = ∂− = = −∂ = −∞ −∞ + ∂∂ = −∞ ⋅ + =
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For 0, ( ) 1 ( ) () 1( ) 1 YX YY X yb a X X a Fy F a f y y F ya f x dx y f aa −∞ <= = ∂−  =   = − = − In general, 1 ( ) || fy f =
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Moments of Functions of RVs Y aX b aX b aX b = + = + = + ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 () VAR Y Y Y a Xb a aXX aX X a VAR X = = +− = = =
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Homework: Uniform is a uniform rv over (0,1).
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This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.

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Function Of One RV - FUNCTIONS OF ONE RANDOM VARIABLE...

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