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 generatedsamples.(a) Note that E ~ Exp(1). What is the probability that P(E < t) for any constantt> 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 densityf(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 commandhist (X, bre aks =50, freqFALSE)

E follows an exponential distribution with parameter value 1. Thus the probability density function…

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?

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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