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STATS 210 random variables

创建日期：2024-04-20 15:40:31

所属分类：统计学statistics

标签： 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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25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 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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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