Assignment05Sol

Assignment05Sol - Columbia University IEOR 4404: Simulation...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Columbia University IEOR 4404: Simulation Fall 2009 Solutions to Assignment 5 1. (Exercise 5.14) (a) Consider X with distribution G . Since the denominator of the frac- tion is equal to G ( b ) - G ( a ), we can easily guess that the infor- mation we have to use in order to write F as a conditional dis- tribution is a X b . Let us prove it. Assume therefore that F ( x ) = P ( X x | a X b ) and prove that F ( x ) = G ( x ) G ( a ) G ( b ) G ( a ) : F ( x ) = P ( X x | a X b ) = P ( X x | a X b ) P ( a X b ) P ( a X b ) = P ( X x, a X b ) P ( a X b ) using Bayes rule = P ( a X x ) P ( a X b ) since x [ a, b ] = G ( x ) - G ( a ) G ( b ) - G ( a ) Therefore, we proved that F can be written as F ( x ) = P ( X x | a X b ). (b) It is important to notice that the distribution F is defined only on the interval [ a, b ]. Therefore, by differentiating F , we obtain the following density: f ( x ) = F ( x ) = g ( x ) G ( b ) - G ( a ) 1 [ a,b ] ( x ) . Now, we have to find an upper bound c for the ratio f ( y ) g ( y ) for all y [ a, b ]. Given the definition of f above, it is easy to see that c = 1 G ( b ) G ( a ) works. Therefore, we are able to apply the usual ac- ceptance/rejection algorithm to generate Y with distribution F : STEP 1. Generate a random variable X with distribution G . STEP 2. Generate
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/19/2011 for the course IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.

Page1 / 6

Assignment05Sol - Columbia University IEOR 4404: Simulation...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online