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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 ~ Erp(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 Erp(1). What is the probability that P(E < t) for any constant 2. t> 0? (b) Given Y = 1, what is the probability that Y will be accepted? (c) What is the joint probability that P(Y < 1,Y is accepted)? (d) Note that the density function f(r) in the samples is the conditional prob. f(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f(x) =D 글리 I>1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)

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 ~ Erp(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 Erp(1). What is the probability that P(E < t) for any constant 2. t> 0? (b) Given Y = 1, what is the probability that Y will be accepted? (c) What is the joint probability that P(Y < 1,Y is accepted)? (d) Note that the density function f(r) in the samples is the conditional prob. f(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f(x) =D 글리 I>1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Could you please help with 3(c), 3(d) and 3(e). Thanks

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 - Erp(1). What is the probability that P(E < t) for any constant t> 0? (b) Given Y = 1, what is the probability that Y will be accepted? (c) What is the joint probability that P(Y < 1,Y is accepted)? (d) Note that the density function f(r) in the samples is the conditional prob. f(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density 12/2 f(r) = I>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 - Erp(1). What is the probability that P(E < t) for any constant t> 0? (b) Given Y = 1, what is the probability that Y will be accepted? (c) What is the joint probability that P(Y < 1,Y is accepted)? (d) Note that the density function f(r) in the samples is the conditional prob. f(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density 12/2 f(r) = I>1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.)
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