class03_8

class03_8 - 4 1 ; 3 / ) 1 ( ) 1 ( ) ( ] 1 [ ] { [ 1 ; 3 2 )...

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1.017/1.010 Class 8 Expectation, Functions of a Random Variable Mean, variance of random variables Expectation (population mean ) of x ... E [ x ] For a discrete x : ) ( ) ( i x i x p x E = = x x For a continuous x : +∞ = = dx x xf E x ) ( ) ( x x (Population) variance of x ... Var [ x ]: () [ ] 2 ) ( x - x x E Var = Functions of a random variable y = g ( x ) x is a random variable with CDF F x ( x ) y is a random variable with CDF F y ( y ) since it depends on x Derived distribution problems Derive F y ( y ) from F x ( x ) using either of these options: 1. Analytical derivation Apply definitions of y , F x ( x ), and F y ( y ). }] ) ( { [ ] ) ( [ ] y x g x P y g P y [ ) ( P y F y = = = = y | x x 2. Stochastic simulation Generate many realizations of x , compute y for each replicate, construct empirical F y ( y ) from y replicates Example (analytical derivation): Distribution of y = x 2 for a uniformly distributed x : Uniform distribution centered on 0: 1
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1 0 ; ) ( ) ( ] [ ] { [ ) ( ] ) ( { [ ] ) ( [ ] [ ) ( 1 1 ; 1 ) ( ; 2 1 ) ( ) ( 5 . 0 5 . 0 5 . 0 5 . 0 5 . 0 2 2 = = = = = = = = = = + = = = y y y F y F y y P y x x P y F y x g x P y g P y P y F x x f x x F x x g y x x y y x x x | x | x x y Uniform distribution centered on 0.5 (note need to split y interval into 2 parts):
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Unformatted text preview: 4 1 ; 3 / ) 1 ( ) 1 ( ) ( ] 1 [ ] { [ 1 ; 3 2 ) ( ) ( ] [ ] { [ ) ( ] ) ( { [ ] ) ( [ ] [ ) ( 2 1 ; 3 / 1 ) ( ; 3 / ) 1 ( ) ( ) ( 5 . 5 . 5 . 2 5 . 5 . 5 . 5 . 5 . 2 2 + = = = = = = = = = = = = = = = + = = = y y F y F y P y x x P y y y F y F y y P y x x P y F y x g x P y g P y P y F x x f x x F x x g y x x x x y y x x x | x x | x | x x y Mean and variance of y = g ( x ) : ) ( ) ( ] ) ( [ ) ( i x i i x p x g g E E = = x y + = = dx x f x g g E E x ) ( ) ( )] ( [ ) ( y y { } 2 ] ) ( ) ( [ )] ( [ ) ( x-x y y g g E g Var Var = = Copyright 2003 Massachusetts Institute of Technology Last modified Oct. 8, 2003 2...
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class03_8 - 4 1 ; 3 / ) 1 ( ) 1 ( ) ( ] 1 [ ] { [ 1 ; 3 2 )...

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