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Stat-461 final exam
Creation date：2024-04-09 17:37:04
Belonging category：数学mathematics
final exam

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Stat-461/Mast-729G/4

Sample questions for final exam
1. Suppose that a store opens at 0pm and customers arrive at the store according
to a non-homogeneous Poisson process {N(t), 0 ≤ t ≤ 8} with the intensity
function
λ(t) =
{
4− t
4
, 0 ≤ t ≤ 4,
3 + t−4
4
, 4 < t ≤ 8.
Let Sj denote the time at which the j-th customer arrives, j ≥ 1. Compute
(a) P [S11 ≤ 5|N(3.5) = 9].
(b) P [S2 ≤ 4|N(6) = 4].
2. Suppose N(t) is a Poisson process with rate function λ(t) = | sin(t)|, t ≥ 0.
Denote the event-times of N(·) by 0 < S1 < S2 < · · · . Find P [S1 ≤ pi < S3],
and generate the first two event-times S1, S2.
3. Let
N = min
{
n ≥ 1 :
∣∣∣∣U21 + · · ·+ U2nn − 0.333
∣∣∣∣ < 0.001} ,
where U1, U2, . . . are iid Uniform (0, 1) random variables. Estimate E(1/N)
by k−1
∑k
i=1(1/Ni) where k ≥ 200 and sd(1/N)/sqrt(k)< 0.004.
4. For independent sequences of iid Uniform (0, 1) random variables {U1, U2, . . .}, {V1, V2, . . .}
define
N = min
{
n : Un ≤ 1
Vn + 1
}
.
Obtain E(N) and joint as well as marginal pdf’s of (UN , VN).
5. (a) Suppose
P (X = i) = 0.2 for i = 1, . . . , 5.
We want to generate random numbers from X by the acceptance-rejection
method using
P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .
for some 0 < p < 1. Find p that minimizes(
max
i
P (X = i)
P (Y = i)
)
and generate a value of X using that p.
1
(b) Suppose
P (X = i) =
i
15
for i = 1, . . . , 5.
We want to generate random numbers from X by the acceptance-rejection
method using
P (Y = i) = p(1− p)i−1 for i = 1, 2, . . . .
for some 0 < p < 1. Find p that minimizes(
max
i
P (X = i)
P (Y = i)
)
and generate a value of X using that p.
6. Let X ∼ Geometric (1/2) and Y ∼ Geometric (1/3) be two independent
random variables. Obtain the conditional distribution of X, given that X =
Y +1, and then generate a value of a random variable having this conditional
distribution.
7. Generate a value of a random variable X with p.d.f.
f(x) = xe−x
2
+
1
2
e−x, x ≥ 0,
using the indicated methods:
(a) Use the acceptance/rejection method, with g(x) = e−x, x ≥ 0, distribu-
tion as g.
(b) Use the composition method.
8. Generate a value of a random vector (X1, X2) with joint pdf
f(x1, x2) = 3(x1 + x2), 0 < x1, 0 < x2, x1 + x2 ≤ 1,
by acceptance-rejection method, using another random vector (Y1, Y2) with
joint pdf
g(y1, y2) =
1
4
√
y1y2
, 0 < yi ≤ 1, i = 1, 2.
What is the expected number of rejections?
9. Suppose (X1, X2) have the joint pdf f(x1, x2) = 2(x1+x2)I(0 ≤ x1 < x2 ≤ 1).
Find the marginal pdf f2(x2) of X2, the conditional pdf f(x1|x2) of X1, given
X2 = x2, and hence, or otherwise, generate a value of (X1, X2).
2
10. Consider a single-server queueing model where customers arrive according to
a homogeneous Poisson process with rate λA = 10. Upon arrival a customer
either enters service if the server is free at that moment or else joins the waiting
queue if the server is busy. When the server completes serving a customer it
then begins serving the customer that had been waiting the longest. If there are
no waiting customers the server remains idle until the next customer arrives.
The service times are i.i.d. Exponential random variables with rate λS = 12,
independent of arrivals.
(a) Generate the arrival times of the first three customers and their departure
times.
(b) Generate the arrival times of the first three customers and their departure
times, given that both Customer-2 and Customer-3 have tolerance times
which are i.i.d. Exponential (15) random variables, i.e., each will leave
after an Exponential (15) amount of time from his/her arrival, if not
served by then.
11. Consider a queue with 2 parallel servers, where server i takes Exponential (i)
amount of time to complete a service, i = 1, 2. Services are independent of one
another and of the arrivals which occur according to a homogeneous Poisson
process with rate 2. Every arrival goes to the idle server if one of the servers
is busy, to server 1 if both servers are free and wait in a common queue if both
servers are busy; the one that has been waiting the longest is served first when
a server becomes idle. Simulate the arrival and departure times of the first 3
arrivals, and calculate the idle times of the two servers until the departure of
the 3rd arrival.
12. Suppose that X has the pmf:
P (X = x) = 3
∫ 1
0
y(1− y)x+1dy, x = 1, 2 . . . .
Generate a value of X by the composition method and also a value of an
antithetic variable X ′ for X.
13. Suppose that (X,Y ) have the joint p.d.f.
f(x, y) =
{
2
√
x, if 0 ≤ x ≤ 1 and 0 < y < √x,
0, otherwise.
We want to use an estimator of the form [Z + c(X − E(X))] to estimate
P{X ≤ 2Y }, where
Z =
{
1, if X ≤ 2Y,
0, otherwise.
3
Find the (theoretical) optimal choice of c.
14. Suppose that (X,Y ) have the joint p.d.f.
f(x, y) = 4e−
x
y
−2y2 , x > 0, y > 0.
We want to estimate θ = P{|X − Y | < 1}. Consider
Z =
{
1, if |X − Y | < 1,
0, otherwise.
(a) Obtain g(Y ) = E(Z|Y ).
(b) Compute
∑2
i=1 g(Yi)/2 by generating Y1 and Y2.
15. Let X = eU , where U is a Uniform (0, 1) random variable.
(a) Compute Var((X +X ′)/2), where X ′ is antithetic to X.
(b) Obtain c that minimizes Var(X + c(U − 0.5)).
16. Suppose that Y is an Exponential (1/2) random variable, and conditional on
Y = y, N is a Poisson random variable with mean y. We want to estimate
p = P{2 ≤ N ≤ 3}. Obtain E(Z|Y ), where
Z =
{
1, if 2 ≤ N ≤ 3,
0, otherwise.
Compute the estimate
∑5
i=1E(Zi|Yi)/5 by generating 5 i.i.d. copies Y1, . . . , Y5
of Y .
Values of Uniform (0, 1) Random Variables
0.81222166 0.05464751 0.06930601 0.94253943 0.79179519
0.49573802 0.96771072 0.89496062 0.89866417 0.85499261
0.67242223 0.33383212 0.08998713 0.67070207 0.32724888
0.24365016 0.14752814 0.57458281 0.82379935 0.48721456
0.37257331 0.22673586 0.22982463 0.96236055 0.87142464
0.64248448 0.15947970 0.31127181 0.37530767 0.46249647
0.22759288 0.61511912 0.42639104 0.00558105 0.57204456
0.78462924 0.83509543 0.67738296 0.99870467 0.29117284
Case category
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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EARN 5 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 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 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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CMP7000A CMP-7038B Math 263 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 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. 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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