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Stat GR5204 function

创建日期：2024-05-18 17:20:50

所属分类：统计学statistics

标签： Stat GR5204

function

Stat GR5204: Practice Final Name (Print):
Fall 2022 Student UNI:
Time Limit: 2 hours 45 minutes Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You
must show your work to get credit for your solutions — correct answers without work
will not be awarded points.
1 10 pts
2 20 pts
3 15 pts
4 15 pts
5 10 pts
6 15 pts
7 15 pts
TOTAL 100 pts
Stat GR5204 Fall 2022 Practice Final - Page 2 of 12 December 05, 2022
1. (10 points) (5 + 5) Suppose that we have a random sample X1, . . . , Xn from a N(0, σ
2)
population.
(i) Find the form of the most powerful test of H0 : σ
2 = σ20 versus H1 : σ
2 = σ21, where
0 < σ0 < σ1.
(ii) For a given value of α, the size of the Type I error, find the explicit form of the
critical value of the above test.
Stat GR5204 Fall 2022 Practice Final - Page 3 of 12 December 05, 2022
2. (20 points) (7 + 3 + 5 + 5) Suppose that we observe data (X,Z) where Z has a Bernoulli
distribution with parameter 1/2 and X|Z has distribution
X|Z ∼
{
N(θ, 100) if Z = 1
N(θ, 1) if Z = 0.
where θ ∈ R is the unknown mean of the two normal distributions.
(a) Construct level 1 − α confidence interval for θ using both (X,Z) (the confidence
interval should depend on Z) [Hint: Consider Z = 0 and Z = 1 separately].
(b) Find the marginal distribution of X.
Stat GR5204 Fall 2022 Practice Final - Page 4 of 12 December 05, 2022
(c) Use only X to test the hypothesis (at level α)
H0 : θ = 0 versus H1 : θ > 0.
Show explicitly how the critical value of the test should be determined. [Note that
the test should NOT depend on Z now.]
(d) Express the power function of this test (as a function of θ) in terms of Φ(·) (the
standard normal c.d.f.).
Stat GR5204 Fall 2022 Practice Final - Page 5 of 12 December 05, 2022
3. (15 points) (5 + 7 + 3) Observations (Xi, Yi), i = 1, . . . , n, are made from a bivariate
normal population with parameters (µX , µY , σ
2
X , σ
2
Y , ρ), and the model
Y = α + βX + ,
is going to be fit to the observed data.
(i) Argue that the hypothesis H0 : β = 0 is true if and only if the hypothesis H0 : ρ = 0
is true.
(ii) Show algebraically that
βˆ
σ˜/
√
Sx
=
√
n− 2 r√
1− r2 ,
where
r =
∑n
i=1(Xi − X¯n)Yi√∑n
i=1(Xi − X¯n)2
∑n
i=1(Yi − Y¯n)2
is the sample correlation coefficient (the MLE of ρ), σ˜2 = 1
n−2
∑n
i=1(Yi− αˆ− βˆXi)2
and S2X =
∑n
i=1(Xi − X¯n)2.
Stat GR5204 Fall 2022 Practice Final - Page 6 of 12 December 05, 2022
(iii) Show how to test H0 : ρ = 0, given only r
2 and n, using t distribution with n − 2
degrees of freedom.
Stat GR5204 Fall 2022 Practice Final - Page 7 of 12 December 05, 2022
4. (15 points) (5 + 5 + 5) Suppose that X1, . . . , Xn are i.i.d Exp(1/µ), where E(X1) =
µ > 0.
(i) Find the mean and variance of X¯n =
∑n
i=1Xi/n. Hence, find the asymptotic
distribution of X¯n (properly standardized).
(ii) Let T = log X¯n. Find the corresponding asymptotic distribution of T (properly
standardized).
(iii) How can the asymptotic distribution of T be used to construct an approximate
(1− α) confidence interval (CI) for µ? Explain your answer and give desired CI.
Stat GR5204 Fall 2022 Practice Final - Page 8 of 12 December 05, 2022
5. (10 points) Let X1, . . . , Xn be a random sample from an exponential distribution with
the density function
f(x|θ) = θ exp(−θx), for x > 0,
and 0 otherwise. Derive the likelihood ratio test of
H0 : θ = θ0 versus θ 6= θ0,
and show that the rejection region is of the form {X¯n exp(−θ0X¯n) ≤ c}, where X¯n is the
sample mean and c is a constant. Show all your steps.
Stat GR5204 Fall 2022 Practice Final - Page 9 of 12 December 05, 2022
6. (15 points) (3 + 4 + 8) Suppose that
Yij ∼ N(θi, σ2), i = 1, . . . , k; j = 1, . . . , ni,
where θi, for i = 1, . . . , k, and σ
2 are unknown.
(a) Find the distribution of Y¯i· = 1ni
∑ni
i=1 Yij.
(b) Suppose that the goal is to estimate τ =
∑k
i=1 aiθi, where ai, for i = 1, . . . , k, are
known constants. Note that τˆ =
∑k
i=1 aiY¯i· is a natural estimator of τ . Find the
distribution of τˆ .
Stat GR5204 Fall 2022 Practice Final - Page 10 of 12 December 05, 2022
(c) Let us consider testing for H0 : τ = 0 versus H1 : τ 6= 0. Describe a test procedure
— the test statistic, its exact distribution (under H0) and the critical value.
Stat GR5204 Fall 2022 Practice Final - Page 11 of 12 December 05, 2022
7. (15 points) (3 + 3 + 4 + 5) The multinomial distribution is a generalization of the
binomial distribution. It models the probability of counts for each side of a k-sided die
rolled n times.

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FE 825 IERG 5130 STAT 8178 EE161 ADS2 G54FUZ CS326 113B ECN 226 QBUS6860 ELEC3202 COMP9334 MGT11B MET CS 777 STA238 00099F Python SMM306 CMT206 EC116 SP MAST90084 CS480/680 BCPM0091 CMPT 300 ECON2209 FY20 STAT 432 CVEN9625 MATH1712 CMP9771M CMP3750M CIS 325 AI506 MAT344H1S MGT7180 PSTAT 160B CSYS5010 1D STAT 453/558 ME152B STA304H1F CSCI-128 ME428 CO3002 FINE 7640 COMP3234 ECSE 223 EE497 – 3D MATH5905 ECS648U ENG5009 MATH10242 EC665 A CPI221 ISOM5220 MACE40402 MGT B456F COMP9315 AFM 113 STAT 778 CS 564 MA231 CS 230 CS 124 COMP5511 CSC317 ITEC1010 ELEC207 INFT 3019 7CCMFM06 EE 161 5G ECLT5810 MATH 7349 CS526 O2 COMP90054 ELE00063M ACCT5034 ARC1015 CSCI 40 CSCI E40 FIN 524B CS 462 389COM CMT309 MFIN703 COMP4250 STA304 STA304H1S STA 1003 FIN 448 BUFN 732 CS520 CVEN3701 ECMT6006 PHAS0100 CSIT314 MDIA 5028 CSCI 2134 MAT 451 CMPT 459 STAT 457 INFO20003 COMP4141 IELE8066 F540 ECM9 CSE 6321 FIN 430 MATH3911 BFC5916 COMP1011 EEE 6209 STATS 240 CW3 COMP206 COMP1003 4331W ISE 530 CMPT 295 MATH 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 CS 333 CP1406 KIT308 Economics 701 ECON3740 ECE 5759 BFC3340 FINC2012 COMP4500 MATH3871 FIT3080 18-819C Economic ECO202Y5Y PHY100 FINC-780 Econ 100B EC602 IE582 EE450 COMP9517 WRDS 150B ECON 570 LECTURE 2 ECA5101 ECE 4420 ADAD9311 FIT3031 CS 544 MF 703 CS 6140 KIT501 ENGSCI 344 CSC171 IE 332 FINM8007 21S2 COSC 1114 FE545 E471 SIT102 CSC108 MAT301 IAE 101 COP3503 STAT2004 EC 303 CS1026 STAT1203 MATH3171 ISOM5160 IFN657 FIT2014 ACCT2102 DSCI550 STATS 330 ELEC421 COMP3670 QBUS6840 MEES:698B MAST30027 MIET1077 Economics IFT 6758 FIT5122 BUSN5101 CptS 111 HRMT5502 Math2018 PP 312 INFS3200 CZ3005 CSSE1001 SENG 2011 MKTG 553Q BUFN 640 COMP507 ETF5952 DATA 311 ECE4043 BANK3011 finance M3H623544 HW1 INFS 7410 AI6103 STAT3888 INFS3208 ACCT7103 ECOS3013 PSYCH 20B F79MA STAT3006 ENGG952 2X MGSC 7000 ECE4132 MSF 526 MATH 3033 CDS 511 EEEE4120 EEET2465 ECA5103 MATH2070 ELEC3104 BU51019 EEE8147 ENGR20003 COSC 1107 ETF3200 FNCE 435 MSFI 435 ISyE6420 GENG4405 HW ETC5523 IE 7315 MOEC0021 STA 106 ENGG 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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