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hw4_solns - ORIE 4580/5580/5581 Solutions for Homework...

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ORIE 4580/5580/5581 Solutions for Homework Assignment 4 Question 1. The c.d.f of X is F ( x ) = x e a 1 ( b - a ) u du = ln( x ) - a b - a for e a x e b . We now compute the inverse of F ( · ): u = ln( x ) - a b - a = ( b - a ) u + a = ln( x ) = x = e a e ( b - a ) u . Thus, F - 1 ( u ) = e a e ( b - a ) u . The algorithm is: 1. Generate U U [0 , 1]. 2. Return X = e a e ( b - a ) u . Question 2. The c.d.f. of X can be computed as follows: F ( x ) = x -∞ λ 2 e - λ | u - c | du = x -∞ λ 2 e - λ ( c - u ) du = 1 2 e - λc x -∞ λ e λu du = = 1 2 e - λc e λu | x -∞ = 1 2 e - λ ( c - x ) for x c x -∞ λ 2 e - λ | u - c | du = c -∞ λ 2 e - λ ( c - u ) du + x c λ 2 e - λ ( u - c ) du = = 1 2 + 1 2 e λc x c λe - λu du = 1 2 - 1 2 e λc e - λu | x c = 1 - 1 2 e - λ ( x - c ) for x > c . We now compute the inverse of F ( · ). For the first “piece,” we get u = 1 2 e - λ ( c - x ) = ln(2 u ) = - λ ( c - x ) = x = c + 1 λ ln(2 u ) . Also, note that x c = ⇒ - λ ( c - x ) 0 = e - λ ( c - x ) 1 = u = 1 2 e - λ ( c - x ) 1 / 2 .
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