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程序代写案例-DRAFT 1 MATH4065

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DRAFT 1 MATH4065
The University of Nottingham
SCHOOL OF MATHEMATICAL SCIENCES
SEMESTER SEMESTER 2021-2022
MATH4065 - STATISTICAL FOUNDATIONS br>Assessed Coursework
Your neat, clearly-legible solutions should be submitted electronically as a pdf file via the MATH4065 Moodle
page by the deadline indicated there. A scan of a handwritten solution is acceptable. Since this work is
assessed, your submissionmust be entirely your ownwork (see theUniversity’s policy onAcademicMisconduct).
Submissions up to five working days late will be subject to a penalty of 5% of the maximum mark per working
day.
IMPORTANT: Your answers should include, where appropriate
(i) any _ code you used;
(ii) any plots you produce;
(iii) explanations of what you are doing and why.
1. For this question you will need the file ”ExperimentData.csv” which can be downloaded from the module
Moodle page. This file contains the results of an experiment in which the input variables (i.e. the predictor
variables) are Type, Group, Input1, Input2 and Input3, and the output variable (i.e. the response variable)
is Output.
(a) Read the data file into _ and use the THQi command to see if there is any graphical evidence for
relationships between Output and the five other variables. [5 marks]
(b) Using the HK command in _, fit a linear model (model 1) in which Output is the response variable and
the five other variables are all included as predictor variables as well as a constant term. Find the
least-squares estimates of the model parameters. Find two of the five other variables which could
sensibly be excluded, and fit a new linear model (model 2) with these variables removed. [10 marks]
(c) Explore how well model 2 fits the data by considering the residuals and looking for evidence that the
residuals are normally-distributed. [5 marks]
MATH4065 Turn Over
DRAFT 2 MATH4065
2. Consider a random sample ೻ႲИ Н И ೻ᅕ from the continuous distribution with probability density functionഃШചЩ Ҳ ྶചᆇႼႲ exp ԕҭചᆇԡ И ച Ҵ ѱИ
where ྶ Ҵ ѱ.
(a) Write down the log-likelihood function ഋШྶЩ. [2 marks]
(b) Use _ to find the maximum likelihood estimate of ྶ given observed data
೻Ⴒ Ҳ ѱМѷѲѱИ ೻Ⴓ Ҳ ѱМѴѱѺИ ೻Ⴔ Ҳ ѱМѶѸѹИ ೻Ⴕ Ҳ ѱМѺѳѲИ ೻Ⴖ Ҳ ѲМѱѵИ೻Ⴗ Ҳ ѱМѹѳѱИ ೻Ⴘ Ҳ ѲМѸѲИ ೻Ⴙ Ҳ ѱМѹѴѵИ ೻Ⴚ Ҳ ѱМѹѶѳИ ೻ႲႱ Ҳ ѲМѲѲ
[5 marks]
(c) Produce a plot of ഋШྶЩ over a suitable range of values with the maximum likelihood estimate of ྶ
shown. [3 marks]
3. Let ೻ and ೼ be independent random variables such that ೻ ۠ ೱШѱИ ѲЩ and ೼ ۠ ೱШѲИ ѳЩ. Let the random
variable ೽ be defined, conditional on ೻, by ೽ ۠ ExpШЦചЦЩ if ೻ Ҳ ച, where ExpШྐྵЩ denotes an exponential
random variable with mean ྐྵႼႲ.
Use simulation to provide numerical answers to the following questions. You will NOT be awarded marks
for any other method.
(a) Find ೨ ԱٺЦЦᅂᅃ ЦЦԽ. [2 marks]
(b) Find ೳШsinШ೻೼Щ Ҵ ѱМѶЩ. [3 marks]
(c) Find ೳШ೻ ҳ ೽Щ. [5 marks]
MATH4065 End

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