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Stat-461 final exam

创建日期：2024-04-09 17:37:04

所属分类：数学mathematics

标签： Stat-461

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

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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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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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