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MATH2801: Theory of Statistics Assignment

创建日期：2024-05-31 11:43:21

所属分类：数学mathematics

标签： MATH2801

Theory of Statistics Assignment

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MATH2801: Theory of Statistics Assignment
Please follow the instructions below for the assignment (worth 15% of the final mark):
Due date: Before 4 PM Tuesday 1st August (Week 10).
Submission details:
1. This assignment may be completed in a group (max. 4 people), or individually.
2. Two Turnitin submission links will be provide in Moodle. Each person in the group
must submit an electronic copy of the assignment via the INDIVIDUAL Turnitin link.
One person must also submit the assignment via the GRADED Turnitin link.
3. The assignment may be typed or handwritten then scanned. It must be submitted
as a pdf file. Please make sure all the names and zIDs of each group member are
written on all pages.
4. Make sure you show all workings. You do not need to include RStudio commands/code.
Try to aim for clarity and conciseness!
Length: At most 8 pages are allowed for your assignment solutions. This cover sheet must be
submitted with your assignment, but it is not counted in the 8-page limit. A single pdf and Word
version of the coversheet is available in Moodle. Please do not exceed the 8-page limit!
Declaration: You must sign and date your submitted assignment, and include the PRINTED
names/zIDs of any other group members below:
I (We) declare that this assessment item is my (our) own work, except where acknowledged, and
has not been submitted for academic credit elsewhere, and acknowledge that the assessor of this
item may, for the purpose of assessing this item:
• Reproduce this assessment item and provide a copy to another member of the University;
and/or,
• Communicate a copy of this assessment item to a plagiarism checking service (which may
then retain a copy of the assessment item on its database for the purpose of future plagiarism
checking).
I (We) certify that I (We) have read and understood the University Rules in respect of Student
Academic Misconduct.
Date:
Name/zID/signature:
Name/zID/signature:
Name/zID/signature:
Name/zID/signature:
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Some of these questions require the use of RStudio. You do not need to include ALL the
RStudio commands/code, only include commands/code of what is asked in the question.
Make sure to include all plots in your assignment.
1. (a) Let X1, . . . , Xn be a random sample of size n from a Poisson distribution with
parameter λ. Find:
i. The moment generating function of W = X1 + · · ·+Xn.
ii. Using the moment generating function of W , calculate E(W ).
Make sure you show all workings for both parts.
(b) Use RStudio to simulate a random sample of size n = 200 from X ∼ Poisson(1.2),
and then create a frequency histogram using these simulated data.
Comment on the shape of the histogram (in no more than 2 lines).
(c) Use RStudio to simulate 10,000 samples of size n = 200 from X ∼ Poisson(1.2).
Then, compute the sample sum Y =
∑n
i=1Xi for each of the 10,000 simulated
samples. Finally, produce a frequency histogram for the sample sums.
(d) Construct a normal quantile plot of these 10,000 sample sums (from part c).
(e) Explain (in no more than 3 lines) whether the Central Limit Theorem is verified
in this simulation study using the plots from parts c and d.
2. Let Y1, . . . , Yn be a random sample of observations from the model
fY (y;ψ) = (ψ + 1)y
ψ, 0 < y < 1, ψ > −1.
(a) Find the method-of-moments estimator ψ˜ of ψ.
(b) i. Find the maximum likelihood estimator ψ̂ of ψ.
ii. Compute the Fisher Information of ψ̂.
iii. Hence, compute the standard error of ψ̂.
(c) Obtain the probability density function of U = − ln(Y ).
3. (a) Why is randomization important in an experiment (explain in no more than 2
lines)?
(b) “Accuracy of the results of a statistical analysis are so important that we should
always take the largest possible sample size”. Comment on the above statement
(in no more than 3 lines).
(c) Suppose we have a population measurement which comes from Normal(µ, σ2)
where µ ∈ R is unknown but σ = 2.5 is known. Find the sample size that will
ensure that the length of a 99% confidence interval for µ is no longer than 1.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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