Assignment05Sol(with_5.26)

Assignment05Sol(with_5.26) - Columbia University IEOR 4404:...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: 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 U ∼ Uniform[0 , 1] STEP 3. If U ≤ f ( X ) c g ( X ) , then set Y=X. Otherwise, go back to STEP 1. By running the algorithm above, we know that Y will have the de- sired distribution F . Nonetheless, the question asks us to show that the following modified version of the usual acceptance/rejection al-...
View Full 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(with_5.26) - Columbia University IEOR 4404:...

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