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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 ~ 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? (b) Given X = x, what is the probability that X can be accepted? (c) What is the joint probability that X is accepted and X = x?

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? (b) Given X = x, what is the probability that X can be accepted? (c) What is the joint probability that X is accepted and X = x?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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help needed until part (e). no need programming for this question.

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? (b) Given X = r, what is the probability that X can be accepted? (c) 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(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f(x) = I > 1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.) (f) Generate 10000 samples according to the algorithm in Part (e), and draw the his- togram plot with command hist (X, bre aks = 50, freq FALSE)
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? (b) Given X = r, what is the probability that X can be accepted? (c) 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(r|accepted). Find f for X, subject to a constant. (e) With the results, write the pseudo-code for the density f(x) = I > 1. (Hint. Find g and h to generate f. For g, you may consider the inversion algorithm.) (f) Generate 10000 samples according to the algorithm in Part (e), and draw the his- togram plot with command hist (X, bre aks = 50, freq FALSE)
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