Đề thi kết thúc học phần học kỳ II năm học 2017-2018 môn Thống kê nhiều chiều giúp các bạn sinh viên có thêm tài liệu để củng cố các kiến thức, ôn tập kiểm tra, thi cuối kỳ. Đây là tài liệu bổ ích để các em ôn luyện và kiểm tra kiến thức tốt, chuẩn bị cho kì thi học kì. Mời các em và các quý thầy cô giáo bộ môn tham khảo.
Trang 1Bai 1 (2 digm)
X1 (ill all C110 X rs-' A.r3(tt, E) v6i X = (X2) ; tt = /22) vil E = (012
(
Tim phan ph6i cfra, voi Yu = X1 — X2 IA Y2 = X2 — X3
I 2
DE THI KkT THOC HQC PH -AN H9c kST II — Nam h9c 2017-2018
MA LLIU TRU
(do ph)ng KT-DBCL ghi)
LP/R TI-126
Ten h9c phan: Themg Ke Nhi6u Chi6u Ma" HP: TT H .2,01
Chi chit: Sinh vier/ dttoc phép 10 kitting dttvc phepl si2 dung tai 1iv khi lam bai
Ghi chü them: SV ducic pile') sir dung mOt td giAy A4 chuAn bi (vi6t tay, có ghi MSSV, h9
ten) va Op kern bai thi
0-12 0-13
022 023
023 033
Bai 2 (2 dim)
Cho X ,, J\/;,(//,, E) (tan OA ma trhn nghich dao E-1)
P.2 E21 E22
voi X1 la, vector Ap (q x 1), X2 lh vector CAP ((p — q) x 1); PI E tz2 E
Tim ham mat di, dieu kien cfra, X1 khi bit X2 = x2
vOi 1 = 1,2, 3 va j = 1,2, , n1 Trong do cac eti dc lap cling phan phi 1V;)(0, E): it lh trung binh
chung; r la anh hrtang cila, lieu phap alit 1 volt E ncri
b Da \Tao bang sau, thilc hien kiem dinh cho nhfing anh hirang cUa cac lieu phap voi mix ST
nghia a = bit rang T44),V = 4,77
De thi Om 3 trang)
1 1/3]
Trang 2iN DE THI KkT THUC HOC PI-1.7^iN
Hoc IcS7 II - Nam hoc 2017-2018
Chung ta tin hanh phan tich thanh phan el-1111h (PCA) tren dti lieu nay sau khi dã chuA,n hoa
10 bien nay Voi cac ket qua, nhan dime (nho phan mem R) dtrdi day, ta nen giff 14i raw thanh phan chit-1h? VI sao? 'At ke cac thanh phan chInh (Noe chpn? Cac bin nao gop phan iOn xay ding nen hai thanh phan el-1Mb dau tien? giti thich (thong qua h s6 Wong quan )?
eigenvalue percentage of variance cumulative percentage of variance comp 1
[1] 2.142118e+00 1.455968e+00 1.149338e+00
[7] 2.163636e-01 1.925230e-01 1.546575e-01
MENA 0.42009993 -0.01555795 -0.315341555 0.19589470 -0.006074950 -0.09766395 ENFA 0.40712005 -0.12264860 -0.072851719 0.96932467 0.277549704 0.57174504
thi g6ni 3 trang) [Trang 2/3]
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(do phong KT-DBCL 9hi)
Rtfd N G DAI HOC KHOA HOC TI.1NHIEN, DHQG-HCM
Ma ji
Hoc kr II — Nam hoc 2017-2018
b) Ngttdi trA, Idi din thoai chi an khi nao du 15 cu6c gi thi moi di an trim Tinh kI \wig thdi gian cho cüa ngtrdi nay
c) Bit rang co k cu6c goi an trong trong bon giO du tiën TInh xac suk de' co j-cu6c gcoi
d6n trong mOt gid du tien (j < k)
b) Tinh so khach trung binh trong cira hang sau Wit thdi gian dai phvc vv
c) Tinh ti I khach hang ti6m nang an ca hang
Cali 5 (2.0 diem)
Cho X(t) va, Y(t) la hai qua trinh Brown chuAn dOc lap Ardi nhau
a) CI-rang minh rang Z(t) = X(t) — Y(t) ding la, qua, trinh Brown
b) Tim phtrong sai cüa Z(t)
A thi gOIT1 2 trang) [Trang 2/2]
Trang 5(H04,, TRU'ONG DAI HOC KHOA HOC NHIEN, DHQG-HCIVI
DE THI KET THUC HOC PHAN Hoc kST II — Nam hoc 2017-2018
MA LU'U TRU
(do plOng KT-DBCL yhi)
C,741/2_171-10
Ten h9c phan: Toan Ting Dung va Thong Ke Ma HP: Ti H
ThOi gian lam bai: 90 phut Ngay thi ,(0/06/ ZO/W
Chi chit: Sin,h, vien ID duct phep / N khong duo'c phepi sz't dung tai lieu khi lam bai
Ghi chit them: SV ditoc phep sit dung mOt to giAy A4 chuL bi sn (vi6t tay, có ghi MSSV, 119 ten) va, Op kern bài thi
BM 1 (1,5 dim) Wit tram chi phat hai loai tin hieu A va, B vOi xac suAt Wong itng 0,84 ye, 0,16
Do có nhi6'u tren cluOng truAn nen 1/6 tin hieu A bi lech va dtroc thu nhlr tin hieu B, con 1/8 tin hieu B bi lech thanh tin hieu A
a Tinh xac suAt thu dttoc tin hieu A
b Giasi thu &roc tin hieu A, tim xac sugt d thu diroc dung tin hieu 1c pita
Bai 2 (2,5 die'rn) Cho vecto ngAu nhien (X, Y) có bang phan phi xac sut sau:
Bai 3 (4 dim) Mt bai bac) trong tap chi Journal of Sound and Vibration (Vol 151, 1991, pp
383-394) mo fa mOt nghien citu v m6i quan h gifia s phoi nhi8m tigng 8n va, viec tang huy6't a
Dir 1iu sau di/0c lgy di din ti chi lieu dttoc trinh bay trong bai bao nay
1 0 1 2 5 1 4 6 2 3
x 60 63 65 70 70 70 80 90 80 80
5 4 6 8 4 5 7 9 7 6
85 89 90 90 90 90 94 100 100 100
a Ve d8 thi phan tan cna y (huy6t áp tinh Wang milimet thity ngan) theo x (cueng clO am thanh
tinh bng decibels) MO hinh hi quy don có phil hop trong trtrang hop nay?
b Ve &rang thing h8i quy tren ding he' truc toa dO ô cau (a) Cho to'ng binh phtfong sai s6
SSE = 31,266, tim ttoc ltrong cna, o-2 (Lroc Wong cho pinking sai caa sai s6 trong m6 hinh hOi
quy don tren)
c Tim mile huy6t áp trung binh Wong ling NT& ctrOng di) am tha.nli 85 decibels Tim khoang tin
cy 95% cho in& huytt áp trung binh nay
d MOt ngtroi cho eang phoi nhi8m titng n va tang huy6t Ai) khOng Wong quan vOi nhau Hay kie'm dinh thuyet tren vói mire Sr nghia 5%
A thi g -trang) rang 1/4)
Trang 6MA LU'U TR e (do pluing K7'-D13CL ght,)
Bai 4 (2 diem) Ta xet mo hinh hOi quy b6i sau: Y =/30 + 01 x1 + 029:2 ± E
a Hohn thhnh bang ANOVA sau:
TOng quat 147889
b Kiem dinh giii thi6t 110 : /31 = 02 = 0 Voi mile Y nghia 5%
HT
(De thi gin 4 trang)
Hp ten ngt1oi ra de/MSCB: Nst.itAp Thi T.g9c Chit kSr [Trang 2/4]
Trang 9‘oc O TRU'ONG DAI HOC KHOA HOC NHIEN, DHQG-HCM
4
Hoc lcST II - Nam hoc 2017-2018
MA LU'U TRU
(do phony KT-DBCL gin)
C.0;11.t MTH M
Ten hoc phan.: LST thuy6t xac suAt cd ban Ma HP: MTH10516
Th6i gian bai: 90 pinit NOy thi • /6(/ Z•01 8
Ghi chit Sinh vien I CI duck phep / khOng duck phepj st't dung tai lieu khi lam bai
• Dt; thi có 28 cau hOi, và co kern theo 2 bang tra cu6i d
Ma d6: 326
• Voi m6i cau h6i, chi có 1 dap an dimg nhAt S dung but chi to kin dap an dttoc ch9n
Bang tra 1Cii:
2 The number of tornadoes in a given year follows a Poisson distribution with mean 3
Calculate the variance of the number of tornadoes in a year given that at least one tornado occurs
3 A delivery service owns two cars that consume 15 and 30 miles per gallon Fuel costs 3 per gallon On any given business day, each car travels a number of miles that is independent of the other and is normally distributed with mean 25 miles and standard deviation 3 miles Calculate the probability that on any given business day, the total fuel cost to the delivery service will be less than 7
(De thi g6nA trang) [Trang
T.16
Trang 10x,oc KH0, 6, TREONG DAI HOC KHOA HOC DHQG-HCM
Calculate the probability that the total benefit paid to the policyholder is 48 or less
Calculate the probability that Actuary Rahul examines fewer policies than Actuary Toby
[j] 0.4000 E 0.3333 E 0.3571 110.3214 El 0.2857
7 A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy To purchase the supplemental policy, an employee must first purchase the basic policy
Let X denote the proportion of employees who purchase the basic policy, and Y the proportion
of employees who purchase the supplemental policy Let X and Y have the joint density function
f(xy) = 2(x + y) on the region where the density is positive
Given that 10% of the employees buy the basic policy, calculate the probability that fewer than 5% buy the supplemental policy
8 An auto insurance policy will pay for damage to both the policyholder's car and the other driver's car in the event that the policyholder is responsible for an accident The size of the
payment for damage to the policyholder's car, X, has a marginal density function of 1 for
0 < x < 1 Given X = x, the size of the payment for damage to the other driver's car, Y, has conditional density of 1 for x < y < x +1
Given that the policyholder is responsible for an accident, calculate the probability that the payment for damage to the other driver's car will be greater than 0.5
9 An insurance company's annual profit is normally distributed with mean 100 and variance 400
Let Z be normally distributed with mean 0 and variance 1 and let F be the cumulative
distri-bution function of Z
Determine the probability that the company's profit in a year is at most 60, given that the profit in the year is positive
(De thi gem g trang)
HQ ten ngttai ra de/MSCB: Nguy'n Van Thin Chit kSr• [Trang 2/g
Trang 11I:1 [F(5) - F(2)]/F(5) E F(2)/F(5)
E 1 - F(2)
10 A car is new at the beginning of a calendar year The time, in years, before the car experiences
its first failure is exponentially distributed with mean 2
Calculate the probability that the car experiences its first failure in the last quarter of some
calendar year
11 The working lifetime, in years, of a particular model of bread maker is normally distributed
with mean 10 and variance 4
Calculate the 12th percentile of the working lifetime, in years
5.30 E 12.35 M 7.65 El 14.70 E 8.41
12 Individuals purchase both collision and liability insurance on their automobiles The value of
the insured's automobile is V Assume the loss L on an automobile claim is a random variable
with cumulative distribution function
F(1) =
3 ( / 17) 0 < / < V -(/ - V)
le V , V <1
10 Calculate the probability that the loss on a randomly selected claim is greater than the value
of the automobile
El 0.25 E 0.10 El 0.75 EJ 0.00 E 0.90
13 An insurance policy will reimburse only one claim per year
For a random policyholder, there is a 20% probability ofno loss in the next year, in which case
the claim amount is 0 If a loss occurs in the next year, the claim amount is normally distributed
with mean 1000 and standard deviation 400
Calculate the median claim amount in the next year for a random policyholder
El moo E 663 I] 873 E 790 El 994
14 Every day, the 30 employees at an auto plant each have probability 0.03 of having one
acci-dent and zero probability of having more than one acciacci-dent Given there was an acciacci-dent, the
probability of it being major is 0.01 All other accidents are minor The numbers and severities
of employee accidents are mutually independent
Let X and Y represent the numbers of major accidents and minor accidents, respectively,
occurring in the plant today
Determine the joint moment generating function Mx,y(s, t)
(De' thi gain ttrang)
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(do phOng KT-DBCL ghi)
TRU'ONG DAT HOC KHOA HOC TV' NHIEN, DHQG-HCM
s
ETQC l(ST II — Nam hoc 2017-2018
15 The random variable X has moment generating function M(t)
Determine which of the following is the moment generating function of some random variable
E ii and iii only
16 A device contains two components The device fails if either component fails The joint density
function of the lifetimes of the components, measured in hours, is f (s, t), where 0 < s < 1 and
17 Let X and Y be the number of hours that a randomly selected person watches movies and
sporting events, respectively, during a three-month period The following information is known
about X and Y:
E(X) = 50, E(Y) = 20, Var(X) = 50, Var(Y) = 30, Cov(X, Y) = 10
The totals of hours that different individuals watch movies and sporting events during the three months are mutually independent
One hundred people are randomly selected and observed for these three months Let T be the
total number of hours that these one hundred people watch movies or sporting events during this three-month period
Approximate the value of PT < 7100]
18 An insurance company insures a large number of drivers Let X be the random variable
representing the company's losses under collision insurance, and let Y represent the company's
losses under liability insurance X and Y have joint density function
2x+42—y, 0 < X < 1 and 0 < y < 2
f(x,Y) = o otherwise
i and iii only
El at most one of i, ii, and iii
Calcul8te the probability that the total company loss is at least 1
(D6 thi gemn g trang)
119 ten ngltdi ra de/MSCB: Nguye-n Van Thin Chit ksr• [Trang44
Trang 13TRUONG DAI HOC KHOA HOC TV NHItN, DHQG-HCM
DE THI KkT THUC HQC PHAN
19 An insurance policy pays for a random loss X subject to a deductible of C, where 0 < C < 1
The loss amount is modeled as a continuous random variable with density function
20 A public health researcher examines the medical records of a group of 937 men who died in
1999 and discovers that 210 of the men died from causes related to heart disease
Moreover, 312 of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men, 102 died from causes related to heart disease
Calculate the probability that a man randomly selected from this group died of causes related
to heart disease, given that neither of his parents suffered from heart disease
21 An insurance agent offers his clients auto insurance, homeowners insurance and renters
insur-ance The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive The profile of the agent's clients is as follows:
i) 17% of the clients have none of these three products
ii) 64% of the clients have auto insurance
iii) Twice as many of the clients have homeowners insurance as have renters insurance iv) 35% of the clients have two of these three products
v) 11% of the clients have homeowners insurance, but not auto insurance
Calculate the percentage of the agent's clients that have both auto and renters insurance
22 In a shipment of 20 packages, 7 packages are damaged The packages are randomly inspected,
one at a time, without replacement, until the fourth damaged package is discovered
Calculate the probability that exactly 12 packages are inspected
23 An auto insurance company insures an automobile worth 15,000 for one year under a policy
with a 1,000 deductible During the policy year there is a 0.04 chance of partial damage to the car and a 0.02 chance of a total loss of the car If there is partial damage to the car, the amount
X of damage (in thousands) follows a distribution with density function
Trang 14KHO, TRU'ONG DAI HOC KHOA HOC TU' NHIEN, DHQG-HCM
<,
TPMOCKIMINH
Hoc 167 II — Nam hoc 2017-2018
24 An investment account earns an annual interest rate R that follows a uniform distribution on
the interval (0.04, 0.08) The value of a 10,000 initial investment in this account after one year
is given by V = 10, 000e'
Let F be the cumulative distribution function of V
Determine F(v) for values of v that satisfy 0 < F(v) < 1
MA LU'U TRO'
(do plant!' KT-.D.I3CL 02)
11 25 [In( v ) 0 0-1
10, 000 • v— 10, 408
25 Let X and Y denote the values of two stocks at the end of a five-year period X is uniformly
distributed on the interval (0, 12) Given X = x, Y is uniformly distributed on the interval
(0, x)
Calculate Cov(X, Y) according to this model
26 An insurance company categorizes its policyholders into three mutually exclusive groups:
high-risk, medium-high-risk, and low-risk An internal study of the company showed that 45% of the
policyholders are low-risk and 35% are medium-risk The probability of death over the next
year, given that a policyholder is high-risk is two times the probability of death of a
medium-risk policyholder The probability of death over the next year, given that a policyholder is
medium-risk is three times the probability of death of a low-risk policyholder The probability
of death of a randomly selected policyholder, over the next year, is 0.009
Calculate the probability of death of a policyholder over the next year, given that the
policy-holder is high-risk
E 0.1215 El 0.0025 II] 0.0200 0.2000 E] 0.3750
27 Each week, a subcommittee of four individuals is formed from among the members of a
com-mittee comprising seven individuals Two subcomcom-mittee members are then assigned to lead the
subcommittee, one as chair and the other as secretary
Calculate the maximum number of consecutive weeks that can elapse without having the
committee contain four individuals who have previously served together with the same
sub-committee chair
28 Six claims are to be randomly selected from a group of thirteen different claims, which includes
two workers compensation claims, four homeowners claims and seven auto claims
Calculate the probability that the six claims selected will include one workers compensation
claim, two homeowners claims and three auto claims
(De' thi Wom g trang)
HQ ten ngt.tdi ra cl6/MSCB: NguyAn Van Thin Chit kSr• [Trang 6/1
Trang 15(I)(z) = P(Z < z) =
•
z 1 e-Pdu 2 -427c
Bang A.2: Phan phOi chug"' tAc
z 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 -3.4 0.0002 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003
*Voi z < -3.50, xat suat so nh6 hon ho4c Wang 0.0002