Question 3 In an investigation into the size of errors produced by two machines, m measurements x = (x1,...,x) are taken from the first machine and n measurements y = (y₁,...,ym) are taken from the second machine. The measurements x=(x1,...,xm) on the first machine can be modelled as a random sample from a normal distribution with known mean μ and unknown precision, T₁ = 12. The measurements y= (y₁ym) on the second machine can be modelled as a random sample from a normal distribution with known mean μ and unknown precision, ₂ = (a) Assuming independent Gamma prior distributions, Gamma (a₁, b₁) and Gamma (a₂, b₂), for T₁ and T2, respectively, what is the joint posterior distribution for t₁ and t₂? (b) What are the marginal posterior distributions for t₁ and T2, respectively? (c) Explain why t₁ and 2 are independent under the posterior. We would like to make inference on d = √√√√ (d) Explain how would you estimate by simulations, the following (i) The posterior mean of d (ii) A 95% equal tail interval for d (iii) How would you estimate P(d < 0)?
Question 3 In an investigation into the size of errors produced by two machines, m measurements x = (x1,...,x) are taken from the first machine and n measurements y = (y₁,...,ym) are taken from the second machine. The measurements x=(x1,...,xm) on the first machine can be modelled as a random sample from a normal distribution with known mean μ and unknown precision, T₁ = 12. The measurements y= (y₁ym) on the second machine can be modelled as a random sample from a normal distribution with known mean μ and unknown precision, ₂ = (a) Assuming independent Gamma prior distributions, Gamma (a₁, b₁) and Gamma (a₂, b₂), for T₁ and T2, respectively, what is the joint posterior distribution for t₁ and t₂? (b) What are the marginal posterior distributions for t₁ and T2, respectively? (c) Explain why t₁ and 2 are independent under the posterior. We would like to make inference on d = √√√√ (d) Explain how would you estimate by simulations, the following (i) The posterior mean of d (ii) A 95% equal tail interval for d (iii) How would you estimate P(d < 0)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Question 3
In an investigation into the size of errors produced by two
machines, m measurements x = (x1,...,x) are taken from the first machine and n
measurements y = (y₁,...,ym) are taken from the second machine. The measurements
x=(x1,...,xm) on the first machine can be modelled as a random sample from a normal
distribution with known mean μ and unknown precision, T₁ = 12. The measurements
y= (y₁ym) on the second machine can be modelled as a random sample from a normal
distribution with known mean μ and unknown precision, ₂ =
(a) Assuming independent Gamma prior distributions, Gamma (a₁, b₁) and Gamma (a₂, b₂),
for T₁ and T2, respectively, what is the joint posterior distribution for t₁ and t₂?
(b) What are the marginal posterior distributions for t₁ and T2, respectively?
(c) Explain why t₁ and 2 are independent under the posterior.
We would like to make inference on d = √√√√
(d) Explain how would you estimate by simulations, the following
(i) The posterior mean of d
(ii) A 95% equal tail interval for d
(iii) How would you estimate P(d < 0)?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 4 images
