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Suppose that g is an easy probability density function to generate from, and h is a non- negative function. Take a close look at the following algorithm pseudo-code: Step 1. Generate Y~ g. Step 2. Generate E ~ Exp(1) in the way that E = -log(U), U ~ Unif (0, 1). Step 3. If E≥ h(Y), set X = Y. Otherwise go to Step 1. Step 4. Return X. This is a rejection algorithm and we want to find the density function of the generated samples. (a) Note that E ~ Exp(1). What is the probability that P(E ≤t) for any constant t> 0? (6) Given X = x, what is the probability that X can be accepted? What is the joint probability that X is accepted and X = x? (d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density C f(x) e-2²12 x > 1. (Hint. Find g and h to generate f. For q, you may consider the inversion algorithm.) =

Suppose that g is an easy probability density function to generate from, and h is a non- negative function. Take a close look at the following algorithm pseudo-code: Step 1. Generate Y~ g. Step 2. Generate E ~ Exp(1) in the way that E = -log(U), U ~ Unif (0, 1). Step 3. If E≥ h(Y), set X = Y. Otherwise go to Step 1. Step 4. Return X. This is a rejection algorithm and we want to find the density function of the generated samples. (a) Note that E ~ Exp(1). What is the probability that P(E ≤t) for any constant t> 0? (6) Given X = x, what is the probability that X can be accepted? What is the joint probability that X is accepted and X = x? (d) Note that the density function f(x) in the samples is the conditional prob. f(x|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density C f(x) e-2²12 x > 1. (Hint. Find g and h to generate f. For q, you may consider the inversion algorithm.) =

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Would like to ask part d) and part e). Already know a) is 1-exp(-t), b) is exp(-h(x)), c) is exp(-h(x))*h(x), please do not answer these 3 parts again. Thank you! 

3. Suppose that g is an easy probability density function to generate from, and h is a non- negative function. Take a close look at the following algorithm pseudo-code: Step 1. Generate Y ~ g. Step 2. Generate E - Exp(1) in the way that E = – log(U), U Unif(0, 1). Step 3. If E > h(Y), set X =Y. Otherwise go to Step 1. Step 4. Return X. This is a rejection algorithm and we want to find the density function of the generated samples. (a) Note that E ~ Exp(1). What is the probability that P(E < t) for any constant t > 0? Given X = x, what is the probability that X can be accepted? fe What is the joint probability that X is accepted and X = x? (d) Note that the density function f(x) in the samples is the conditional prob. f(x| epted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f (x) = e2, x > 1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
Transcribed Image Text:3. Suppose that g is an easy probability density function to generate from, and h is a non- negative function. Take a close look at the following algorithm pseudo-code: Step 1. Generate Y ~ g. Step 2. Generate E - Exp(1) in the way that E = – log(U), U Unif(0, 1). Step 3. If E > h(Y), set X =Y. Otherwise go to Step 1. Step 4. Return X. This is a rejection algorithm and we want to find the density function of the generated samples. (a) Note that E ~ Exp(1). What is the probability that P(E < t) for any constant t > 0? Given X = x, what is the probability that X can be accepted? fe What is the joint probability that X is accepted and X = x? (d) Note that the density function f(x) in the samples is the conditional prob. f(x| epted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f (x) = e2, x > 1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
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