Considering the rejection method algorithm, all the required conditions are satisfied with the candidate density
𝑔
(
𝑥
)
and multiplicative constant M, what is the range of
𝑓
(
𝑥
)
𝑀
𝑔
(
𝑥
)
?

Question

Considering the rejection method algorithm, all the required conditions are satisfied with the candidate density 
𝑔
(
𝑥
)
 and multiplicative constant M,  what is the range of 
𝑓
(
𝑥
)
𝑀
𝑔
(
𝑥
)
?

BlackTom AI · Accepted Answer

First, recall the rejection sampling setup: there is a target density f(x) that we want to sample from, a proposal density g(x), and a constant M such that f(x) ≤ M g(x) for all x in the domain. Therefore, the ratio r(x) = f(x) / (M g(x)) satisfies 0 ≤ r(x) ≤ 1 for all x, provided f and g are nonnegative densities and M is chosen to satisfy the inequality. Now, evaluate the implications for the range of r(x).

- At points where f(x) = 0 (which can occur for many x in a density with finite support or where the target density assigns zero probability), r(x) = 0.
- At points where f(x) attains the maximum allowed by the bound f(x) ≤ M g(x), we have f(x) = M g(x) and thus r(x) = 1.
- For any other x where f(x) is strictly less than M g(x), r(x) lies strictly between 0 and 1.

Thus, the complete range of the ratio f(x) / (M g(x)) is the closed interval [0, 1], including both endpoints. The proposed option (0, 1) is missing the endpoints and therefore is not entirely correct. The endpoints matter because there exist x where the ratio can be exactly 0 (when f(x) = 0) and where it can be exactly 1 (when f(x) = M g(x)).

In summary, the ratio f(x) / (M g(x)) lies within 0 and 1 inclusive, i.e., the interval [0, 1].

25F-STATS-102C-LEC-1 Quiz3- Requires Respondus LockDown Browser

Considering the rejection method algorithm, all the required conditions are satisfied with the candidate density 𝑔 ( 𝑥 ) and multiplicative constant M, what is the range of 𝑓 ( 𝑥 ) 𝑀 𝑔 ( 𝑥 ) ?

View Explanation

Log in for full answers

Similar Questions

Let 𝑓 ( 𝑥 ) = 𝑐 𝑜 𝑠 ( 𝑥 ) , for 𝑥 ∈ [ 0 , 𝜋 / 2 ] . Suppose we want to sample 𝑓 ( 𝑥 ) through the rejection method, with 𝑈 𝑛 𝑖 𝑓 ( 0 , 𝜋 / 2 ) as the candidate density. What is the multiplicative constant M?

Which of the following descriptions is correct about the rejection method for sampling from a target density 𝑓 ( 𝑥 ) ?

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

More Practical Tools for International Students

Homework AI Solver

Stylized Writing Assistant

Plagiarism Checker Assistant

Citation Assistant

In-Class Translation Assistant

Class Notes Assistant

Quiz Search Assistant

Past Exam Questions by School

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler