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Assignment 2

This document outlines the instructions and questions for STAT 2006B/C Assignment 2, due on February 20, 2025. It includes requirements for submission format, handwritten solutions, and significant figures for numerical answers. The assignment consists of five questions related to statistical concepts in nuclear physics, confidence intervals, maximum likelihood estimation, and Bayesian statistics.

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15 views2 pages

Assignment 2

This document outlines the instructions and questions for STAT 2006B/C Assignment 2, due on February 20, 2025. It includes requirements for submission format, handwritten solutions, and significant figures for numerical answers. The assignment consists of five questions related to statistical concepts in nuclear physics, confidence intervals, maximum likelihood estimation, and Bayesian statistics.

Uploaded by

0715 hky
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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STAT 2006B/C Assignment 2

Due date & time: February 20, 2025, at 23:59

Instruction:

1. File Format: Your assignment must be submitted in PDF format. Only one PDF
file should be submitted for the entire assignment.

2. Handwritten Solutions: All solutions must be handwritten. Please write clearly. Each
solution must include all steps leading to your final answer.

3. Numerical Answers: All numerical answers must be presented in 3 significant figures.

Note: you may apply the pdf formula of the order statistics from the lecture notes.

Question 1

In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a
detector, particles of known energy are directed into it. The values of signals from 15 different
detectors, for the same energy, are

260 216 259 206 265 284 291 229 232 250 225 242 240 252 236

(a) Find a 95% confidence interval for 𝜇 , assuming that these are observations from a 𝑁(𝜇, 𝜎 2 )
distribution.

(b) Construct a qq-plot of the data. Is the normality assumption in (a) reasonable?

Question 2

Let 𝑋̅ be the mean of a random sample of size 𝑛 from 𝑁(𝜇, 9) . Find the smallest 𝑛 so that
𝑃(𝑋̅ − 1 < 𝜇 < 𝑋̅ + 1) ≥ 0.90.

Question 3

Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from a Gamma distribution with the probability density
function given by
1 2 −𝑥/𝜃
𝑓(𝑥; 𝜃) = 𝑥 𝑒 , 0 < 𝑥 < ∞, 0 < 𝜃 < ∞.
2𝜃 3
(a) Find the maximum likelihood estimator 𝜃̂.

(b) Find the mean and the variance of the Gamma distribution using the moment generation
function technique.

(c) Find the Rao-Cramér lower bound, and thus the asymptotic variance of the maximum likelihood
estimator.

(d) Is the maximum likelihood estimator 𝜃̂ efficient?

Question 4

Let 𝑋1 , 𝑋2 , … , 𝑋𝑛 be a random sample from a distribution with pdf 𝑓(𝑥; 𝜃) = 𝜃𝑥 𝜃−1 , 0 < 𝑥 < 1 ,
where 0 < 𝜃.

(a) Find a sufficient statistic 𝑌 for 𝜃.


(b) Show that the maximum likelihood estimator 𝜃̂ is a function of Y.

(c) Argue that 𝜃̂ is also sufficient for 𝜃.

Question 5

Let 𝑋1 , … , 𝑋𝑛 be a random sample from a Poisson distribution with mean 𝜃. Let the prior pdf of 𝜃 be
gamma with parameters 𝛼 and 𝛽. The pdf of the gamma distribution is

𝜃 𝛼−1 𝑒 −𝜃/𝛽
ℎ(𝜃) = , 0 < 𝜃 < ∞.
Γ(𝛼)𝛽𝛼

(a) Find the posterior pdf of 𝜃, given that 𝑋1 = 𝑥1 , … , 𝑋𝑛 = 𝑥𝑛 .

(b) The mean of the posterior pdf of 𝜃 by the moment generating function technique. Hence, find a
Bayesian point estimate of 𝜃.

(c) Show that the point estimate found in (b) is a weighted average of the maximum likelihood
estimate 𝑥̅ and the prior mean 𝛼𝛽, with respective weights of 𝑛/(𝑛 + 1/𝛽) and (1/𝛽)/(𝑛 + 1/𝛽).

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