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STATS 210 random variables

Creation date：2024-04-20 15:40:31

Belonging category：统计学statistics

Tag STATS 210

random variables

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STATS 210

1. Suppose that the random variables X and Y have the following joint pdf:
fX,Y (x, y) =

2 for 0 < x < 1 and 0 < y < 1− x
0 otherwise
Answer the following questions:
(a) Find the marginal pdf of Y . For writing down your result, replace . . . with the correct
expression:
fY (y) =
{
. . . for 0 < y < 1
0 otherwise
[3 marks]
(b) Show that the pdf of X given Y has the following expression for 0 < y < 1:
fX|Y (x|y) =

1
1− y for 0 < x < 1− y
0 otherwise
[1 mark]
(c) Use the result obtained in part (b) to show that
E(X|Y = y) = 1− y
2
for 0 < y < 1
[3 marks]
(d) Employ the result obtained in part (c) to write down the expression of E(X|Y ) and
then use this expression for calculating the ratio
Cov(X,Y )
Var(Y )
without computing Cov(X,Y ) and Var(Y ). Assume that 0 < Var(Y ) <∞.
[3 marks]
(e) Decide if the random variables X and Y are independent (or not). Justify your answer.
[1 mark]
[Total: 11 marks]
1
STATS 210
2. Suppose that each of two statisticians A and B must estimate a certain parameter θ whose
value is unknown (θ > 0).
Statistician A can observe the value of a random variable X, which has the Poisson
distribution with mean 2θ. The value observed by statistician A is X = 3.
(a) Write down the likelihood function for this problem, LA(θ; 3). Remember to state the
range of values of θ for which the likelihood is defined.
Find the maximum likelihood estimate of θ. In your answer, you should differentiate
the likelihood with respect to the parameter θ, and set to 0 for the maximum. Quote
your numerical answer to 1 decimal place.
You should make reference to the graph of the likelihood function, which is shown in
Figure 1. Show all your working in your answer. [5 marks]
0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
L A
(;
3)
Figure 1: Plot for question 2, part (a).
Question 2 continues on the next page
2
STATS 210
Statistician B can observe the value of a random variable Y ∼ Gamma(3, θ). Suppose
that the value observed by statistician B is Y = 2.
(b) Write down the log-likelihood function for this problem, logLB(θ; 2). Remember to
state the range of values of θ for which the log-likelihood is defined.
Find the maximum likelihood estimate of θ. In your answer, you should differentiate
the log-likelihood with respect to the parameter θ, and set to 0 for the maximum.
Quote your numerical answer to 1 decimal place.
You should make reference to the graph of the log-likelihood function, which is shown
in Figure 2. Show all your working in your answer. [5 marks]
0 1 2 3 4 5
-14
-12
-10
-8
-6
-4
-2
0
lo
gL
B(
;2
)
Figure 2: Plot for question 2, part (b).
(c) Use the results obtained in part (a) and in part (b) to show that there exists a constant
C (which does not depend on θ) such that
logLB(θ; 2) = logLA(θ; 3) + C for all θ > 0.
Find the numerical value of C. Quote your answer to 4 decimal places. [2 marks]
(d) Compare the values of the maximum likelihood estimates that you have obtained in
part (a) and in part (b).
Use the identity from part (c) in order to justify why the maximum likelihood estimate
obtained by statistician B should be the same as the maximum likelihood estimate
obtained by statistician A. Explain your answer. [5 marks]
[Total: 17 marks]
3
STATS 210
3. Suppose that either Instrument A or Instrument B might be used for making a certain
measurement. From inspecting visually the instrument, we cannot distinguish between
Instrument A and Instrument B.
Instrument A yields a measurement whose (marginal) pdf is:
fX(x) =

2x for 0 < x < 1
0 otherwise
Instrument B yields a measurement whose (marginal) pdf is:
fY (y) =

3y2 for 0 < y < 1
0 otherwise
(a) Find the (marginal) cdf of X. For writing down your result, replace . . . with the correct
expressions:
FX(x) =

· · · for x ≤ 0
· · · for 0 < x < 1
· · · for x ≥ 1
[4 marks]
(b) Find the (marginal) cdf of Y . For writing down your result, replace . . . with the correct
expressions:
FY (y) =

. . . for y ≤ 0
. . . for 0 < y < 1
. . . for y ≥ 1
[4 marks]
Suppose that one of the two instruments is chosen uniformly at random and a measure-
ment T is made with it. Note that T is a continuous random variable.
Let I be a Bernoulli random variable with the property that I = 0 if Instrument A is
chosen and I = 1 if Instrument B is chosen.
It is evident that P(I = 0) = P(I = 1) = 1/2.
(c) Assume that 0 < t < 1. Fill in the gaps indicated with . . . in the following identity:
P(T ≤ t) = P(T ≤ t| . . .)P(. . .) + P(T ≤ t| . . .)P(. . .).
[2 marks]
Question 3 continues on the next page
4
STATS 210
(d) Use the results obtained in parts (a)-(c) in order to show that the expression of the
(marginal) cdf for T is:
FT (t) =

0 for t ≤ 0
t2 + t3
2
for 0 < t < 1
1 for t ≥ 1
[3 marks]
(e) Find the (marginal) pdf of T . For writing down your result, replace . . . with the correct
expressions:
fT (t) =
{
. . . for 0 < t < 1
. . . otherwise
[3 marks]
(f) Assume that 0 < t < 1. Fill in the gaps indicated with . . . in the following identity:
P(I = 1 |T = t) = f...|...(t | I = 1)P(. . .)
f...(. . .)
Name the theorem that is the basis for the identity above. [3 marks]
(g) Show that P (I = 0 |T = 1/4) = 8/11. [3 marks]
[Total: 22 marks]
5
STATS 210
4. Let U ∼ Uniform(0, 1). We define T = log U
1− U = log(U)− log(1− U).
Answer the following questions:
(a) Verify that the function g(u) = log(u) − log(1 − u) is (strictly) monotone increasing
by showing that
dg(u)
du
> 0 for 0 < u < 1. [1 mark]
(b) Show that lim
u→0
g(u) = −∞ and lim
u→1
g(u) =∞. [1 mark]
(c) Find the pdf of T , remembering to include the range of values of t for which your
answer is valid. [3 marks]
[Total: 5 marks]
5. Suppose that the distribution of (X,Y ) is Bivariate Normal, the marginal distribution
of X is Normal(0, 1) and the marginal distribution of Y is Normal(0, 9). The correlation
between X and Y is ρX,Y = −1
3
.
We define:
V = X,
W = −kX + Y,
where k is a constant.
Answer the following questions:
(a) Name the joint distribution of (V,W ). Do not use calculus. [1 mark]
(b) Write down the expression of Cov(V,W ). [2 marks]
(c) Find the value of k such that the random variables V and W are independent.
[2 marks]

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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 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LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 FINC5090 MATH1052 SciComp Elec4622 FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 MAT 237 BISM2201 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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum

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