A part is selected at random and found to be defective.. b What is the probability that thesecond one selected is defective giverr that the lth on" was defective?. ' c What is the proba
Trang 1Nhu vfy, qua kh6u lya chgn biiSn ta dgOc
Hinh 5.9 PhAn du chuin ho6 theo quan sat c0a s6 tieu dQ tan
Ki6m tra phan du cta m6 hinh ndy: Ching h4n, theo chi s5 i ta th6y c6
2 gi|triphAn du chu6n ho6 (tmg vdi quan s6t thri 6 vi thrl l0) vuqt qu6 2; vi
ph4m thir hai ld d; khA nhd t4i c6c quan s6t ll *24 Dir sao hai vi ph4m niy
cirng kh6ng a6n ndi nio Pfr6n du chuAn ho6 x6p theo x1, x2 hay ! ddu kh6ng
c6 vi ph4m tl6ng kC Ta lga chqn (*) ldm m6 hinh cu5i cirng #
a) Define a random experiment that
involves the lifetime ofthe pC
b) What is the sample space?
c) Define two events that are disjoint.
d) Define two events that have a
nonempty intersection
1.2* Customers access an AutomatedTeller Machine (ATM) They want
to withdraw random amounts of
money in multiples of 50 thousand.
VND Specify the sample space Is
it the discrete one? Specift threeevents of interest
lraP
$6.1.8Ar TAP CHTI0NG r
1.1 Tu6i thg cria mQt chi6c pC tlu-o.c
tfnh tir hic n6 bit ttAu ho4t riQngttrin khi h6ng
a) Xdc dinh ttri nghigm ngiu ntridngin v6i tu6ithg cria pC.
b) Kh6ng gian miu 6 d6y td gi?c) X6c ttlnh 2 bi6n c6 xung khfc.d) X6c tlinh 2 birin c6 c6 giao khrictrting
D& b)S = IR* = (0; -);
c) (0;1000) vi (>2000);
d) (0;1000) and (900;2000).1.2* Circ khrich hing lui t6i mqt chiiicm6y rrit tidn tu rlQng Hq muiin rrit
mQt lugng ti6n ngSu nhi6n 50 ngdntl6ng mQt Hiiy chi 16 kh6ng gian
m6u Edy phrii chnng ld kh6nggian mdu roi r4c? Chi ra 3 bi6n ci5
quan tim.
D,S S = {50,1 00, ,1 04} ;Dfrng; (< t03), (t03; 5.103),15.103+t04)
Trang 21.3** Consider the random experiment
of tossing a single die once and
courrting the number of dots facing
tup Assume that P({6})=0.3 and
all other faces are equiprobable
Find the probability of the events
A={2,4,6}, B={1,5},
6 = {1 2 3, 4} ancl D= Au(BnC)
1.4 Let P(A; = 0.9; P(B) = 0.8 Show
that P(A n e) > o.z .
1.5* Given that P(A) = 0.9:
1.7** Consider tlre switching network
It is equally likely that a switch
will or will not work Find the
prob that a closed path will exist
between terminals A and B
L3** xdt thi nghiQm ngiu nhi6n tungcon xirc xic don I litr vi di5m s6
diu ch6m hi€n tr6n mdt Gie sfr
P({6})=0,3 vd t6t ci c6c m[t kh6c li ddng khn ndng Tim x6c
1.7** Xdt mOt mach tliQn nhu hinh v€
Cdc c6ng tic it6ng hoic md v6ikhi nlng nhu nhau Tim x6c su6tel6 c6 it ra mQt dudng d5n gita 2diu n5i A vi B
D& 0,688
234
1.8 We place at random n particles in
m > n boxes Find the probability p
that the particles will be found in npreselected boxes (one in each
box) Consider the following cases:
c) F-D (Ferrni-Dirac) - the particlescannot be distinguished, at mostone particle is allowed in a box
1.9* A random experiment has samplespace S={a,b,c} Suppose that
P{a,c} = 0.75 and P{b,c}=0.6
Find the probabilities of the
elementary events
l.l0** lf m students born onindependent days in lg93 areadterrding a lecture Find theprobabilitiy that at least two of
' them slrare a birthday arid showthat p>l ,2 *n"nm=23.
1.8 Chtng ta d4t ng6u nhi6n n h4t
(ph6n t&) viro m > n hQp Tirn x6csu6t p cl€ c6c h4t du-o c tim th6y d n hQp chqn tru6c (m5i hpt chi d trong
DS a) "' : b),n,-, (m+n_l)1,n!(m - n)!
g) ]-l- .
'm!
1.9* MOt thf nghiQm ngiu nhi€n c6kh6ng gian miu S={a,b,c} Gi6
srlr P{a,c} =0,'15, P{b,c}=0,6. Tim xic suSt cria c6c bitin cti socap
oS e(a) = O,+; P(b) =0,25;P(c) = 9,35'
ft0** Gi6 sri c6 m sinh vi€n sinh nrmt993 ttang tham dU gio giAng Timx6c su6t itnhit2 sinh vi6n cirng c6ngiy sinh vd chftng t6 ring p >1
Trang 3l.l1** A train and a bus arrive at the
station at random between 9 A.M
and l0 A.M The train stops for l0
rninutes and the bus for a minutes
l.ind a so that the probability that
the bus and the train will meet
equals 0.5
1.12 A fair die is rolled two tirnes Firrd
the probability that the sum of dots
is 7
l.l3* We have two coins; the first is
fair and the second two-headed
We pick one of the coins at random,
vv€ toSS it nvice and heads shows
both times Find the probability
that the coin picked is fair
1.14 Show that n(nln) defined by
Eq (1.2.1) satisfies the three
axioms of a probability, that is:
1.ll** Tiu ho6 vd xe bus tdi ga t4i m6t
thoi di,3m ngSu nhi6n tir 9 tl6n l0gid Tdu dirng trong l0 phrit cdn xebus dirng a phrit Tim a tl6 xric su6t
xe kh6ch vi tdu ho6 g{p nhaubing 0,5
D.s 60-J100 phirt
1.12 Tung con xric xic cdn A5i Z tin.
Tim x6c sudt aii t6ng s5 n6t bing 7.
oS: l.
6
1.13* C6 2 tt6ng tiBn, mQt cdn d6i, mQtc6 2 mf;t s6p Rtit ngiu nhi6n I
d6ng tidn, tung n6 2 lin vd d6uhign m4t s6p Tim xdc su6t ddngtirSn rtt duo c ld ddng ti6n cdn d6i
PS: l.
5
1.I4 Chring t6 ring e(ele) theo(1.2.1) thod mdn 3 ti€n d6 cta xdcsu6t, d6 ld:
ay e(nln)>o;
uy n(sle)= r ;c; P(A,r-,rerln)=
a) Find the probability of the event
that two faces are the same withoutthe information given
b) Find the probability of the sameevent with tlre information given
1.18*" lwo manufacturing plantsproduce sirnilar parts Plant I
produces 1000 parts, 100 of whichare defective Plant 2 produces
2000 parts, 150 of which aredefective A part is selected at
random and found to be defective
What is the prob that it came fromplant I ?
1.19** A lot of 100 semiconductor chipscontains 20 that are defective Two
clrip.s are selected at random, without
replacement, frorn the lot
a) phat is the probability that thefirst one,selected is defective?
b) What is the probability that thesecond one selected is defective
giverr that the lth on" was defective?
' c) What is the probability that bothare defective?
1.20*,' Box I contains 1000 bulbs of which l0 percent are defective
Box 2 contains 2000 bulbs of
\
1.17** Xet thi nghiQm tung 2 con xirc
xic c6n aOi Si6t ring tting c6c n5t
l6n hon 3
a) Tim x6c su6t biiin c6 2 mdtgi6ng nhau khi kh6ng biiit th6ngtin dE n6u
b) Tim x6c su6t cfra bii5n c6 tr€nv6i th6ng tin dl cho
ps, 1,1.
6'33
\
1.18** Hai nhi m6y s6n xudt nhirng
linh kiQn giting nhau Nhd m6y Isin xu6t 1000 linh kiQn, trong d6c6 100 li h6ng Nhd m6y 2 sanxu6t 2ooo linh ki€n, trong d6 c6
150 la h6ng Chgn ng5u nhi6n I
' linh kiQn vi th6y ring n6 bi h6ng.
Tim x6c suSt n6 do nhi m6y I sinxu6t
D& 0,4
.V
1.19** L6 hdng 100 chip bdn din c6chira 20 clrip b! h6ng Chgn ngiunhi6n 2 chi6c kh6ng l{p lai.a) X5c su6t chitic thri nhSt b! h6ng
li bao nhi6u?
b) X6c su5t chii5c thti 2 b! h6ng li
bao nhi6u, bii5t ring chii5c thfr nh6t
2000 trong d6 5% bi
Trang 4which 5 percent are defective Two
bulbs are picked from a randomly
selected box
a) Find the probability both bulbs
are defective
b) Assuming that both are defective,
find the probability that they came
fi'om box l:
c) Than find the probability that
the next bulb picked from the
selected box will be defective
1.21** SLrppose that laboratory test to
detect a certain disease has the
fol lowin g statisiicq Let
' 4 : leveni that the tested person
iras the disease)
B * {event that the test result is
positive) It is known that
e(ela) = o.ee; r(ale) = o.oo5
and 0.1 percent the population
actually has the disease
What is the probability that a
person has the disease given that
the test restrlt is positive?
1,22** We have two cages of
experimental mice In the first
there are l0 male and 15 female
ones; in the second there are 8 and
7, respectively Catch one mouse
frorn the first cage and one mouqe
from the second cage then 6rass
them to the third cage Finally
catch one mouse from the third
cage Find the probability that this
mouse is male
' Hai b6ng duo c rirt ra t& mQt h6p
du-o c chgn ng6u nhi6n
a) Tim x6c su6t ci hai b6ng ttdu bi
thu dugc kiSt qui sau ddy:
n(nla) = o,ee; n(n[) = o,oo5
vdi A = {ngudi kiiim tra c6 bQnh},
B : {krit qui ki6m tra duorg tinh}
vit 0,1 %o ddn si5 bi bQnh niy.
Tinh x6c su6t m6t ngudi bi bQnh,
bi6t rang kiSt qui ki6m tra li duongtinh
D,S 0,165
ry
1.22t't' C6 2 l6ng chuQt thi nghiQm,
I6ng thrl nh6t c6 l0 con chuQt dgc
vi 15 con chuQt c6i; l6ng thti II c6
8 con chuQt tllrc vd 7 con chuQt cdi
Bit I con tu l6ng I, mQt con tirl6ng II rdi dua sang ldng III; saud6 bit I con tir l6ng III Tinh x6csu6t d.l con niy ld chugt <lgc.
DS:0,467
communication channel The
channel input symbol X may:rssume the state 0 or the state l.
Because of the channel noise, an
irrput 0 may convert to an output Iand vice versa
The channel is characterized by the
channel transition probabilityPo,Qo,Pl, and ql, definedbY
oo = e(lr l*o), p, = R(volxr),
oo =e(volxs), andql =R(vrlxr),here xs and x1 denote the events(X =0) and (X = l), respectively,and y6 and y1 denote the events(Y = 0) and (Y = l), respectively
Note that po + qo = I = Pl * q1 Let
P(xo) : 0.5, po : 0.l, and pt = 0.2
a) Find P(ye) and P(y1)
b) If a 0 was observed at theoutput, what is the probability that
a 0 was the input state?
c) lf a I was observed at theoutput what is the probability that
a I was the input state?
d) Calculate the probability of error P
1.23* Xet k€nh th6ng tin nh! ph6n DAu
viro X cfia k6nh du-o c xem nhu 6 2trang th6i 0 ho4c l Do c6 nhi6uk6nh truydn, diu ra 0 c6 thii ring
v6i dAu viro I vi ngucr.c l4i K€nhdugc tl{c truorg bdi x6c su6t
b) Ni5u thdy 0 d diu ra, xac suit tI6 0
(de) la tr4ng thrii cfia dAu vdo?
c) NiSu th5y 1 d diu ra, x6c su6t d6
I (de) Ia tr4ng th6i cta tliu vdo?d) Tfnh x6c su6t sai lim P .
DS a) 0,55;0,45; b) 0,818;
c) 0,889; d) 0,1 5
Trang 51.24**., The incidence of an illness in
the general population is q A new
medical procedure has been shown
to be effective in the early
detection of tlre illness The
probability that the test corlcctly
identifies someone without ihe
illness as negative is 0.95
a) Find the probability that
someone with the illness will get
positive result
b) It's given that's q =0.0001 and
the prpbability that the test
correctly identifies someone with
the illness as positive is 0.99 You
take tlre test and the result is
positive What is the probability
tlrat you have the illness?
c) Now it's known that q = Q.2
and someone with positive result
Find the probability that someone
positive result
Find the probability that the test is
correct
1.25 How marry equations do you need
' to check in order to establish
independence of 5 events?
1.26 Let g
= [0; l] x [0; t] Assume that
P(A) is equal to the area of A Find
t\Yo independent events A, B that
rlo not have a rectangular form
1.27** A systcnr consisting ofseparah:
componcnts is said to be a parallel
t.24** Tf lQ mic mQt losi bQnh criacQng d6ng ld q Quy trinh mdi t6 rahigu qui d6 ph6t hipn s6m benhndy X6c su6t x6t nglriQm chi th!
ihing m6t nguoi khrSng bi bQnh c6k6t qud 6m tinh ld 0,95
a) Tim x6c suAt phin ring duongtinh c0a ngudi kh6ng c6 bQnh.
b) Bitit ring q=0,0001 vi xdc
su6t xdt nghiQm chi th! tfiing mgt
ngudi b! bQnh c6 k6t qui duo:rgtinh ld P=0,99 B4n thgc hiQn
x6t nghiQm cho k6t quA duongtfnh Khe ndng b4n b! mic b€nh lnbao nhi€u?
c) Biiit ring q = 0,2 vi x6c sudt ddmgt ngudi khi c6 k6t qui duongtinh s€ bi b€nh ld 0,8 Tim x6c sudtphin trng duong tinh cria ngudi c6b€nh
Tim x6c su6t xet nghigm tfiing
ES a) 0,05; b) 0,165;
c) 0,80; 0,92
1.25 Bao nhi€u phuong trinh ban cAna6 ttrii5t l$p tinh rtQc t6p cria 5 biiSnc6z
D& 65
1.26 Gia sfr S=[0; I] x [0; l] Cho rangP(A) bing diQn tich A Tim 2 biiin
cO aQc lgp A, B mi kh6ng c6 d4ng_o chB nhflt
1.27*J' MQt he th6ng c6c thdnh phin
ri6ng rE xem nhu mQt h€ song song
system if it functions when at leastone of the components functions
Assume that the components fail
independently and that theprobability of failure of component
i is pi, i- 1,2 ,n Flnd the
probability that the system
functions
1.28 Let S be the sample space of
an experiment and S={A,B,C} ,P(A)=p, P(B)=q, and P(C)=r, where p,q,r>0 The experiment
is repeated infinitely, and it isassumed that the successive
experiments are independent
Find the probability that the event
A occurs at least once after the nthexperiment and than firrd theprobability of the event tlrat A
occurs before B
1.291' Manufacturer sells 20000
products, 300 of which are
defective Distributor tests at
random 100 produces, ifthere is at
, rrost one fault, the products will be
accepted Find the probability that
the batch ofgood is refused
n6u n6 ho4t tlQng khi it nh6t rnQtthinh phin hoat tlQng Gi6 sir c6cthinh phin h6ng h6c m$t c6ch tlQc
lfp vi xic su6t h6ng cira thinh phin thri' i ld p;, i=1,2, ,n.'fim
x6c sudt it6 h€ hoat dQng
It
os l-flni.
i=t
1.28 Gia sri S li kh6ng gian m6u c6c
thi nghiQm vd S={A,B,C}, P(A)=p, R(B)=e, P(C)=r,
vdi p,q,rr9 Lap lai thf nghiQut
vd h4n lAn vi gii sri ring c6c thinghiQm thinh c6ng ln dQc lflp.Tim x6c su6t tt€ bii5n cti A x6y ra it
nh6t I IAn sat thi nghiQm thri n r6isau d6 tim x6c su6t cira bi6n c6 A
xiy ra tru6c bitin c6 B
D,si I ; P(A)/[P(A)+ n(e)].
n:
1.29* Nha sin xu6t b6n ra 20000 sinphAm, trong tt6 c6 300 phi5 phim
Nhi phin ptrOi t<i6m tra ngiu nhi€n
100 bO, n6u c6 kh6ng qu6 mQt b0t6i ttri ctr6p nh6n 16 hing Tinh x6c
su6t 16 hnng bi tir chi5i
DS:0,443
Trang 62.1** Two basketballers one by one
throw a ball into a basket until
there is one ball thrown into the
basket The probability of success
in every throwing is 0.8 and 0.6 for
the fist anfthe second basketballer,
respectively Find the probability
2.2* A group 4 children of the ages
from 0 to 3 Two children are chosen
at random then their ages are added
together We call X the result Find
the probability table of X
2.3* A personnel officer has a number
of serious candidates'to fill four
positions The chance for every
candidate is 0.6 Find the
probability that he has to consider
7 persons correctly
2.4** An information source generates
symbols at random from a
four-letter alphabet {a, b, c, d} with
probabilities P(a)= 1,
P(b)=i,
I
P(c)= P(d)=r A coding
scheme encodes these symbols into
binary codes as follows:
,tnem Tim x6c su6t cia:
a) 56lin ndm cta cAu thri thri nh6t;
bi 56 lin n6m cria c6 2 ciu tht.
D,S a) P{X = n} = 10,08)n-r 0,92;
b) P{Y = 2n - 1} = 0, 08n-1 0,8;
P{Y = 2n} = 0,08n-1 0,12.
2.2*.Mqlt nh6m tr6 gdm 4 em tu6ittr 0tti5n 3 Chgn ngiu nhi6n hai em r6i
:.
cQng hai tu6i cria chtng l4i v6inhau Ggi X ln kiit qui Tim ph6nb6 x6c su6t cia X.
2.3* Cin bQ phdng nhdn sg c6 mQt lo4tring vi6n tldng tlon vdo 4 ch6tilSng Khi nlng thdnh c6ng criam5i ung vi6n li 0,6 Tim xic suStd,5 c6n b0 d6 phni xem x6t ding
lugc tl6 m6 mi ho6 c6c kf hiQu nny
thinh mE nhi phin nhu sau:
2.5* Consider the experiment of throwing a dart onto a circularplate with unit radius Let X be the
RV representing the distance of thepoint where the dart lands from theorigin of the plate Assume that thedart always lands on the plate andthat the dart is equally likely to
land anywhere on the plate FindP(X <a), P(a <X <b) (a <b<l).
2.6* a) Verif, that the function p(x)defined by
2.7** Consider a functionr(x) =1fe-1x2+x-a), -@ < x < @
Gqi X le BNN kf hiQu ttQ dii cria
m6, tt6 ln s5 4i hi€u nhi ph6n (s6bit) Tap gi6 tri cia X li gi? Gi6 sr!
viQc sinh kj hi$u li ttQc lflp, tfnh
c6c x6c su6t R1X=l), P(X=2),
P(X =3) vi P(X >3).
D,s {t,z,sl; }, l, l o
2.5* Xdt thi nghiQm n6m phi ti6u vio
mQt c6i tlTa hinh trdn b6n kinh tlsn
vi Gqi X ln birin ngiu nhi€n chikhoing c6ch tir tli6m phi ti6u cham
vio dia t0i tam cria dTa Gii sri phi
ti6u lu6n roi vio dia vi ch4m vio' -.t
mgi tli6m cria tlia v6i.khi nlng nhu
nhau TimP(X < a), P(a <X < b), (a < b < l).
li hdm khlii'luqng xlc suSt (pnifl cta BNN rdirac X nio tl6
r(x)=
uf e-("2***u)' -@ <.x <co .
243
Trang 7Find the value of a such that (x) is
a probability density function (pdf)
2.10* The number of telephone calls
arriving at a switchboard during
any l0-minute period is known to
be a Poisson RV X with l = 2
a) Find the probability that more
than three calls will arrive during
any l0-minute period
b) Find the probability that no calls
will arrive during any lO-minute
period
c) Calculate E[X], VXl, Mod(X)
2.11'* A production line manufactures
1000-ohrn (C)) resistors that have
l0 percerrt tolerance Let X denote
the resistance of a resistor
Tim gi6 tri cria a sao cho f(x) ln
him rnflt d0 (pdfl cria BNN li6n
tgc X.
D& a=1.
4
2.8 BNN X dugc gqi ld c6 phdn b6Rayleigh niiu him m6t d0 cria n6
cho bdi
rx (x) =f e-*2rtzo2)u1*;
a) Tim him ph6n b6 n*1x1
b) Ve Fy(x) vd fy(x) khi o = l
2.9** X6t BNN chuin X v6i cric tham
b) Tim x6c su6t kh6ng c6 cugc ggi
ttiSn nio trong vdng l0 phfit
c) Tinh ElXl, VlXl, Mod(X)
D& a) 0,143; b) 0,135
Y2.11** MQt d6y chuy6n sin xuit diQn
trd 1000 6m (O) ttusc phdp x6dich l0% Kf hiQu X la tr! sti criattiQn trd Gi6 sri X c6 phdn b6
Assuming that X is a normal RV
witlr mean 1000 and variance
2500 Find the probability that aresistor picked at random will berejected
2.12**, In thq manufacturing ofcomputer memory chips, company
A produces one defective chip forevery nine good chips Let X be
time to failure (in months) of chips It is known that X is anexponential RV with parameter
I
L=; for a defective chip and
^", /"=- with a good chip, Find the
probability that a chip purchased
randomly will fail before:
a) Six months of use;
b) One year of use.
2.13 The radiat miss distance [in meters
(m)l of the landing point of aparachuting sky diver from the
center of the target area is known
to be a Rayleigh RV X with
'l'parameter o- = 100
a) Find the probability that the skydiver will land within a radius of
I 0 m frorn the center of the target
afea,b) Find the radius r such that the
' probability that X > r is e-l .
2.14r"'.lt is known that the floppy disksproduced by company A will bedefective with probability 0.01
The company sells the disks in
chuin v6i trung binh 1000 vd
phuong sai 2500 Tim x6c sudt m$tchi6c di€n trd chgn ng6u nhi6n bilo4ib6
)_ ES 0,045.
2.12*r, Trong viQc s6n xu6t chip nh6
mdy tinh, c6ng ry- A sin su6t I
chi6c h6ng v6i cd 9 chiric t6t Gie
sir X li tu6i thg (theo th6ng) cria
c6c chip Bi6t ring X ti BNN mfiv6i tham ,o i,=] aoi voi ctrip
Z
h6ng " vd ,t = + l0 v6i chi6c chip t6t.Tim x6c su6t di5 I chi6c duoc chgnngdu nhi6n sE bi h6ng:
a) Sau sdu th6ng sri dUng;
b) MQt nIm srl dgng
DS; a) 0,501; b) 0,729
2.13 D0 lQch (theo m6t) cria di6m tiiSp
d6t cta vfn rlgng vi€n nh6y dir t6itdm virng mgc ti6u li BNN X c6ph6n b6 Rayleigh X v6i tham s6
o-= 100
a) Tim x6c su6t tlii vQn clQng vidnnhdy dir ti€p d6t trong vdng b6nkfnh r : lOm tri tdm virng mgc
I0 chii5c mQt vdi lli eldm bdo li s6
Trang 8packages of l0 and offers a
guarantee of replacement that at
most I of the l0 disks is defective
Find the probability that a package
pr.rrchased will have to be replaced
2.15 Let a RV X denote the outcome of
throwing a fair die Find the mean
(expected value) and variance ofX.
2.16* Let X be an exponential RV with
parameter i' Verifr that,
r-_l
E[X]=: -i'7"' and V[X]=
Find Med(X) Which is larger,
E[X] or Med(x)?
2.17t,,' Consider a sequence of
Bernoulli trials with probability p
of success This sequence is
observed until the first' success
occurs Let the RV X denote the
trial number on which this first
success occurs Then the
probability mass function (pmf) of
X is given by
px (k) = P(X = k)
= (t - p)o-' p, k = 1,2,
(There must be k - 1 failures before
the first success occurs on trial X)
The RV X is called a geometric
2.15 Gqi X le BNN chi kiSt qui khi rrlt
mQt con xirc xic cdn <li5i Tim gi6
tri trung binh (ki vgng) vi phuong
sai cria X
DS:3-5: "12 I.
2.16* Gqi X le BNN ph6n b6 mfr thams6 I ri6m tra ring,
I r- IElXl =; vi V[X
L t- L,'
Tinh Med(X), so s6nh v6iE[X].
x
2.17t"' X6t dny c6c phdp tht Bemoullivdi x6c su6t thinh c6ng p Ddy ndy
duo c quan sit tt6n ldn thri thinhc6ng tliu ti€n Gii sir BNN X kf
hiQu sii lin th& cho tltin ph6p tht
thinh c6ng ttiu ti€n Khi tl6, him kh6i luqng x6c su6t (pmf) cta X
cho bdi
px (k) = P(X = k)
=(t-p)u-'p, k=1,2,
(Phni c6 k - I th6t bai trudc lin thri
thinh c6ng X iliu ti6n) BNN Xttugc ggi le BNN c6 ph6n tO trintrhqc v6i tham sti p
a) Chung t6 ring py(k) thoi mdn
q
phuongtrinh fRx(k)=I.
k=l
b) Tim him phdn bO r*1x;.
b) Find the cdf Fy(x) of X.
c) Find the 'expected value E[X]
and the variance V[X].
2.18** Consider an exponential RV X with parameter 1 Show that the
RV X haS the memorylessproperty, that is: For all c, d > 0,P[X >c+dlx >d]= P(X >c) -
2.19 Let the pdf of RV X be given by
fy(x) = kxe-xu(x).
a) Find the constant k, Mod[X].
b) Finct E[x] Elx2l, vtxl.
c) Find the pdf ofthe RV lE.
2.20 Find the mean and variance of a
Rayleigh RV (see Prob.2.8)
2.21** Given that X is a Poisson RVand Px (0) = 0.0498 , compute E[X]
and P(X > 3)
2,22 A RV X is the Pareto random
variable with parameter a, b(a, b > 0) if its pdf is given by
z.t{.*.xdt BNN mii X v6i tham si5 l.
Chi ra reng BNN X c6 tinh ch6tkh6ng c6 tri nhq chinh ld: V6i mgic,d>0,
P[X>c+dlx>d]=P(X>c).
2.19 Hdm mit ttQ cta BNN X cho bdify(x) = kxe-xu(x).
a) Tim hang s6 k, Mod[X].
b) Tirn E[x], ElX2l, VlXl.
c) Tim him m6t tl$ cria 1&.
DS 3; 0,5767
2.22 BNN X ld BNN Pareto vdi c6ctham s6 a, b (a,b>O) n6u himmAt dg cria n6 cho bdi
fx(x) =(a / b) (b/x)a+l, x e[b; o)
247
Trang 9a) Show that E[Xn] exists if and
' onlyifn<a.
b) Find E[X] and Elx2l (a> 2)
2.23* Show that for a Cauchy RV with
the parameters a b with pdf
fy(x1=1 O- - xeR
n(x-a)'+b'
(b > 0) tlre mean does not exist
2.24 l-et X - N(pr, o2 ) , evatuate E[X3 ]
2.25 Suppose.that Z - N(0,1) .
a) Evaluate ElZl.
b) Show that
E[22"7= 1.3 (2n - l)
2.26)'*.ln the orange-region the number
of oranges in a tree follows normal
distribution ln the 600 counted
trees there are 15 ones with less
than 20 orallges and 30 ones with
less than 25 oranges
a) Estimate the average oranges in
one tree
b) Estimate the ratio of the trees
with more than 60 oranges
2.27** (Total Probability Theorem for
Expected Value) Consider a RV X
on a sample space S Consider a
partitiorr {81, ,8n} of S Define
' a) Chi ra rang E[Xn] t6n tai n6u
vdchiniiu n<a.
b)Tim E[X] vd Elx2l (a>2).
2.23* Chi ra ring a6i vOi BNN Cauchy
b) Chi ra ring E[22")= 1.3 (2n -1).
/
2,.26** O mQt vtng trdng cam, nguditath6y s6 qui cam tr€n mQt cdy tu6n
theo ph6n bl5 chuAn Nguli ta d6mthfi 600 cAy thi th6y 15 c6y c6 it hsn 20 qui, 30 cdy c6 it hon 25que
a) Hey udc tugng si5 qui cam trungbinh tr6n mgt cdy
b) U'6c lusng ty IQ cdy c6 tir 60 gL
qua trd l6n
DS a) 5l,l;b)29 %
2.27* (Dinh lf x6c sulit toin phAn vdi
k) vgng) Xdt BNN X tr€n kh6nggian m6u S Xdt ph6p phdn ho4ch{B1, ,Bn} cta S Dat
nrimutes of red What is expected
delay in the journey if you arrive atthe junction at a random time
tuniforrnly distributed over the
whole 5:minute cycle?
Hint: X - the delay, T - the time
when you arrive at the junction
\ 0<T<2: X=0;
2<T<5: X=5-T'
Use tlre Problen 2.27
2.31 Suppose a warship takes l0 shots
at a target, and it takes at least 4
9o
Elxl=fe1x>r<1.
k=02.2g Giesri X li BNN Poison vdi thams6 i tim Mod[X] khi ].>l vd
khi l.<1.
D,S 0<1"<l : Mod[X]=0; t.> l:
uoarxt={lhl, }'"e{2,3,"'l
Lf-l tr6i lai2.30** Chu k! c0a tldn hiQu giao th6ngg6m 2 phfrt xanh r6i d6n 3 ph0t rt6.Tinh thoi gian cho trung binh cria
chuy6n di n6u ban tti5n ngd tu t4im$t thdi ili€m ngiu nhi6n ph0n b6
tl6u trong khodng thdi gian 5 phrit.HD: X- Thdi gian chd, T - thoidirlm b4n d6n ngd tu
Trang 10hits to sink it If the warship has a
record of hitting with 20% of its
shots in the long run, what is the
clrance of sinking the target?
2.32** Wearever tires have a truct
record of lasting 56000 miles on
everege, with a standard derivation
of 8000 miles, and a normal
distribution
a) What is tlre chance that a given
tire will last 50000 miles?
b) Wlrat is the chance that all four
Wearever tire on my car will last
50000 rniles?
' ph6t trring ai5 Aanir chim n6 Trong
mQt thli gian ddi, nguli ta ghi
nhgn ring c6 20% lAn tiu bin
trring dich, co hQi di5 b6n chim mpcti€u Ii bao nhi€u?
,s 0,1209.
K
2,32"* Nhiing chi6c l6p cria hdng
Wearever de ghi nhfn mQt k! lgckhring khiiip h tli tlugc trung binh
56000 d{m v6i rtQ l6ch chuin 8000d[rn vi c6 ph6n b6 chuin
a) Tfnh xric su6t a,i t cni6c l6p decho di dugc it ra 50000 dam
b) Tinh x6c su6t tlti cA 4 chitic liipWearever tr6n xe cria t6i s6 tlidugc ft ra 50000 d4m
$6.3 8Ar TaP CHUONG rrr
random vector (X, Y)?
3.2 Suppose we select one 'point
at
random from within the circle with
radius R If we let the center of the
circle denote the origin and define
X and Y to be the coordinates of
the point chosen, then (X, y) is a
uniform random vector with the
prob density function (pdf) given by
3.2 Gie sri ta chgn I di6m ng5u nhidntrong hinh trdn b6n kinh R N6u kf
hiQu tdm vdng trdn ld gi5c to4 116 vi
X vd Y li to4 tlQ c0a diiSm chgn,
khi d6 (X,D Ie VTNN vdi hdmmft ttQ x6c su6t (pdf) cho b&i
r*" 'rr\ "' 1*,y1 = {k' *2 + Y2 < R2
[o' *2+Y2'R2
a) Determine the value of k
b) Find the probability that thedistance from the origin of the pointselected is not greater than a < R.
c) Find the marginalpdf of X, y.
3.3** A manufacturer has been using
processes to make computer
memory chips Let (X,y) be arandom vector, where X denotesthe time to failure of chips made
by process A and y denotes the
time to failure of chipsmade by
3.4* Let (X, Y) be a random vector,
where X is a uniform RV over(0; 0.2) and Y is an exponential
RV with parameter 5, and X and y
are independent
a) Find the pdf of (X, y).
, b) Find P(Y < X)
a) X6c ttinh gi6 tri cria k
b) Tim x6c su6t mA khoing c6ch tirg5c d6n di6m chgn kh6ng vugt qu6
<t6n h6ng cria chip san xudt b6i-quy
trinh B Gid st hdm mflt d0 cria
(x,Y) ta
f (*,y) = {ub"-(u**u'), x > o,y > o
[0, nguoc laitrong d6 a=10-4, b=1,2.10-4,tinh F(X > Y)
0<x<0,2, y>0nguo c l4r
251
Trang 113.5 Let the pdf (X, Y) be given by
a) Find the marginal cdf of X and Y
b) Show that X and Y are
independent
c) Find
P(x < l,Y < t), P(x > x,Y > y).
3.7 Consider the birrary communication
channel as in Prob 1.22 Let
(X, Y) be a random vector, where
X is the input to the channel and Y
is the output ofthe channel Let
35 Cia sft m{t ttQ cira (x, Y) cho bdi
fxv (*,y) = {*t-*(v*t)' x > o'Y > o
[0, ngu-d c l?i
a) Chring t6 ring fly(x,y) tho&
min phuong trinh
3.7 Xdt k6nh th6ng tin nh! ph6n nhu &
bni tfp 1.22 Gia sri (X, Y) lAVTNN trong d6 X ln diu vio k6nh
vd Y li dAu ra cria k6nh Gii sfr
r(x=0)=9,5t
R(x=o)=6.5,
P(v = r;x = o)= o.l, and P(Y=olX=t)=0.2.
a) Find the probability mass
function (pmf) of (X, Y)
b) Find the marginal pmf of Xand Y
c) Are X and Y independent?
3.8** The pdf of a random vector
(X, Y) is given by
f(x-v)=lo(**Y)' \ "' lo, .otherwiseo 1x'Y 12'
wherekisaconstant.
a) Find the value of k
b) Find the marginal pdf of Xand Y
c) Are X and Y independent?
3.9* Let (X,Y) be a random vector
a)pyy(0,0) = 0,45; pay(0,1) = 0,05;pxv (1,0) = 0, l; p1y(1, l) = 0,4
b)px(0) =px(l) = 0,5;
Pv(o) = 0,55;py(l) = 0,45.c) Dfine
\a3.8** Hdm phdn b6 cfia (X,Y) cho bdi
r;
ul{(*+l)14, -' 0<x<2.
10, nguo.clai 'c) Kh6ng
3.9* Cho (X,U HVTNN Chungt6rdng
[E1xv1]2 s e(x'?)e(v'?) r.i,
tting thfc (BDT) niy c6 t6n li
BDT Cauchy-Schwarz
Trang 123.10** The pdf of a random vector
a) Find the value of k
b) Are X and Y independent?
3.11** Consider the random vector
(X, Y) of Prob 3.8
r1a (vlx) and raly (.lv) .
3.13 Suppose that a random vector
(X,Y) is uniformly distributed over
a unit circle (Prob 3.2)
a) Are X and Y indepenA"i,tt
b) Are X arrd Y correlated?
N
3J0** Hdm mQt <10 cria VTNN (X, Y)cho bdi
a) Tim gi6 tri k
b) Compute the conditional means
E(ylx) ano e(xlv).
3.15 Let (X,Y) be a normal random
vector Determine E(Ylx).
3.16 The pdf of a random vector (X, Y)
is given byfxy (*,y) = {t-(**')' x > o,Y > o
|.0, otherwise
a) Are X and Y independent?
b) Find the conditional pdf of X
3.17 Let (X1, ,Xn) be an n-variatenormal random vector with its pdfgiven by Eq (3.6.a) Show that if
, the covariance of X; and X; iszero for i * j, that is,
' (t
LU ,* J,
then X1, ,Xn are independent
3.14 Gia st (X, y) te VTNN v6i him
kh6ng tuong quan v6i i * j, tr?c li cov(x;,x.;)=E,j={"i i=i
[o i*i,
thi X1, ,X1 ErtQc l{p,
Trang 133.19 Let X and Y be two independent
identically normally distributed
RVs with zero mean and variance
4 Find the probabilities that the
random pirint (X,Y) belongs to the
circle witlr center at (0,0) and
3.22*r' Let (X, Y) be a normal random
vector with covariance matrix I.
cirng phdn b5 chu6n vdi trung binh
0 vi phuong sai 4 Tim x6c su6t d6di,5m ngiu nhi6n (X,Y) thuQc viohinh trdn tfrm t4i (0, 0) vi b6n kinh
D,S.'a, b
v'3.22r'* Gii st (X, Y).ld VTNN chuinv6i ma trfln tuong quan f, .
3.26* If X is N(0;2),y=3X2, findElYl, VIYI, fy(y) and Fv(y).
9.27),t Let y = x2 Find the pdf of y
(see Prob 2.19) and Y= arctanX,
then'Y is uniform in the interval
(-n12,-xl2).
b) N6u z=(o l\
\r 9J' titn he t'
tuong quan gita X vi Y
3.23 N6u X - U(0;l), tim hhm m6t d0cria Y=aX+b, a,be IR.
3.26* Niiu x - rrr(o;z), y =3X2,, tim
ElYl, VlYl, fy(y) vd ev(v).
3.28 Gia st Y=tanX Tim him mdt
<lQ cta Y n6u X - U(-rc / 2; x I Z)D,Sj BNN Cauchy vdi tham si5 l.
3.29 Chung t6 ring n6u BNN X c6 mft
ilQ Cauchy v6i a=0; b=l (xembni tflp 2.19) vi Y= arctanx, khi
d6 Y phin b6 ddu tr6n
(-n12,-nl2).
Trang 143.30 Let X be a continuous RV with the
| ^-xpdr ry (.) =
t;,
^'
::l
Find the transformation Y = g(X)
such that the pdfofY is
3.32 The a-centered clipper is described
' by the following transformation
3.33** Suppose that (X,Y,Z) is a
normal random vector with mean
' vector p = (0,1,2)T and covariance
( t o.s o.s\
tt
matrix E=|0.5 2 0 l Find
the pdf of the RV T = X-2Y +32
and calculate pxy, pyz.
j.sg** Gii sri ring (x,Y,z) le VTNN
p = (0,1,2)T vi rira t{n tuctng quan
( t 0,5 0,5)
tt
mat d0 cria BNN T =X-2Y +32
vA tinh pry, pyz.
S.SO Cia sfr X li BNN li6n tpc vdi mft
3.34 Consider Z: X + Y Show that if
X and Y are independent PoissonRVs with parameters ?y and ?,2,respectively, then Z is also aPoisson RV with parameter 1.1 + 2u2.
3.35 Suppose that X and Y are
independent standard normal RVs
vd hem m$t dq d6ng thli f*" (*,y) DAt z =max(x,v) .
a) Tim him phin b6 ciaZ.
b) Tim him m4t tlO cria Z niSu X vi
Y tlQc lflp
DS; a) Fz(x) = F1v(x,x);
b) f2(x) = fx(x)Fv(x) + Fy (x)fy(x)3.39 MOt hi€u tli€n thii V ld hi.rn cria
' thdi giair t vi cho bdi
Trang 15v (t) = X cos ort + Y sin <ot
in which co is a constant angular
frequency and' X=V-N(O;o2)
and they are independent
a) Show that V(t) may be written
as V(t)=Rcos(o:t-@).
b) Find the pdf of RVs R and show
that R and @ are independent
Determine the joint pdf of
Y;,Y2 and Y3
3.42 Write down the pdf of the random
vector variable
(x,Y) - N(0, 6, 4,9, -0.1) .
3.43 Let X and Y be defined by
f, = cos@, Y = sin @, where @ is
a' random variable uniformly
distributed over (0; 2zr)
' v(t) = Xcosort + Ysinrot
trong d6 c,: li tAn sii g6c kh6ng d6i,
X = Y - N(o;"'z) vd chong tlQct0p
a) Chrng t6 rdng c6 th6 vitit V1t)du6i dangV(t) = Rcos(ot -O).
b) Tim him ph6n bti cria BNN R
vd chring t6 ring R vd @ ttQc lap
E^Si b) fe(r) =;|e '2 tt2o2) ,(r > o)
fs6(r,0) = fn(r[o(0) .
3.40* Ci6 srl X1, ,X1 li n BNN tt$c
lfp ctrng ph6n b6 v6i hdm mat ttQ
(x) Ci6 sri W = Min(X,, .,Xn).
Tim hdm m4t tIO cia W
f" (y) = p"-pru (y).
Find the densities of the following
3.47 The RV X is of discrete type
, taking tlre values xn withP{X = xn} = pn and the RV y is of
continuous type and independent
of X Show that if Z=X+ Y and
W = XY, then
a) X vA Y khdng tuong quan,'b) X vi Y kh6ng tlQc lap
D,Si a) E[XY]=ElXl ElYl;
b) Elx2y2l + E1x2l Ely2l 3.44 a) Hdm g(x) tlon rliQu tIng vi
Y = g(X) Chring t6 ring
.r*.rr)=fFx ,r ' Lr" (x), (v), ndu ndu y v < >e(x); g(x).
b) Tim Fx" (*,y) ntiu g(x) ttcntliQu gi6m
3.45 Cdc BNN X vi Y dQc l{p v6i mflt
d0 mfi
fx (*)=o"-"*u(*);
fy (y) = pe-Pru (v).
Tim him m{t tlQ cta cic BNN:
a)2X+Y; b) X-Y; c) X/Y;
d) Max(X,Y); e) Min(X,Y).
3.46 Chring t6 ring:
a) T(ch chfp cia 2 mdt tlQ chuAn ldmat dO chuin;
Trang 163.49* Let Y = sin X, where X is
uniforrnly distributed over (0; 2z)
Find the pdf of Y.
3.50 Consider an experiment of tossing
a fair coin 1000 times Find the
probability of obtaining more that
520 heads:
a) By using formula (3.7.10);
b) By formul a (3.7 12)
3.51 The number of cars entering a
parking lot is Poisson distributed
with a rate of 100 cars per hour
Find the time required for more
than 200 cars to lrave entered the
parking lot with probability 0.90:
3.48 Gia s& X li BNN v6i him mSt ttQ
f*(*).Gii srl Y=lxl Tim him
3.49* Cho Y = sin X, trong tl6 X c6
phdn b6 ddu tr6n (0;2n) Tim himmat d0 cta Y.
D^Si
f"(y)= lr",[4), -l<v<l
10, nguo.c lai
3.50 Xet thi nghiQm tung d6ng ti6n c6n
d5i looo dn Tim x6c su6t nh6n
xe tr6n gid Tim thoi gian cAn thiiit
d6 c6 qui 200 xe vio bdi tt{u v6i
x6c su6t 0,90:
a) Dtng cdng thric (3.7.6);
b) Dirng c6ng thircP(Y < x)
^v <D((x + I / 2 -?i t Jt')
DS: a) 2,1 89h; b) 2,195h
s6.4.BAr raP CHTIONG rV
4.1* The "cold start ignition time" of an
automobile engine is beinginvestigated by a gasolinemanufacturer The following times(in seconds).were obtained for a testvehicle: 1.70, 1.92, 2.62, 2.45, 3.09,3.15,2.53,and l.19
a) Calculate the sample mean andsample standard deviation
b) Construct a box plot of the data
4.2 A second formulation of the gasoline
was tested in the same vehicle, with
the followirrg tirnes (in seconds):
I.80, 1.99, 3.13, 3.29, 2.75, 2.87,3.40,2.46, 1.89 and 3.35 Use thisnew data along with the cold starttimes reported in Exercise 4.1 to
construct comparative box plots
Write an interpretation of the
inforrnation that you see in these
plots
4.3 Suppose that we have a samplex1, ,x,1 and we have calculated
Xn and Sfr for the sample Now an
(n + l)u' observation becomesavailable Let fn*1 and Sl*1 be
the sample mean and samplevariance for the sample using all
' n*lobservations.
computed using fn and xn*1
b) Show that nSl*, =(n - t)Sl +(n(n + l))(xn*1 -Xr)2.
4.1* HSng san xu6t khi de nghi€n ctu
thdi gian d6nh hla kh0i <lQng l4nh
cta dQng co 610 Khi thti vdi mQt .!.
chi6c xe tdi, ngu&i ta thu duo c c5cs6 tieu sau (tlon v!: gi6y): 1,70;
1,92; 2,62; 2,45; 3,09:, 3,15; 2,53;1,19
a) Tinh trung binh miu vi ttQ l$ch
chuAn m6u
b) Xay dpng d6 th! dang hinh h$p
cta d& liQu
dung so ll9u nay cung vor so lr9u
d bdi t$p tr6n vd thbi gian xu6tph6t l4nh tt6 xiy dung dd thi sos6nh d4ng hinh h$p ViiSt mO ta vd
th6ng tin mi bpn th6y duo c ttr c6c
aO tni nay
43 Gia sri ta c6 m6u x1, ,xn vi ta
dE tinh duo c Xn vi 5l cho miu niy B6y gid ta c6 tiiip quan s6t
th& n + l Gi6 sri Xnal vi Sl*, Utrung binh miu vi phuong sai
m6u hiQu chinh cho m6u s& dgng
tit ci n + I quan s6t 'a) Xn+l c6 th6 duo c tinh nhu th6
ndo khi sri dung Xn vd xn*1?b) Chi ra ring nSfr*, =(n - r)Sl +(n(n + l)[xn*1 -Xn)2.
Trang 174.4* Suppose we have a random
sample of size 2n from a
population denoted by X, and
E(X)=p and V(x) =o2 L"t
be two estimators of p Which is
the better estimator of p? Explain
your choice
4.5** Let Xt, X2, , X6 denote a
random sample from a population
having mean p and variance o2.
Consider the following estimators
a) Is either estimator unbiased?
b) Wrich estimator is best? In
what sense is best?
4.6* Data on pull-off force (pounds) for
connectors used in an automobile
engine application are as follows:
79.3, 75.1, 78.2, 74.1, 73.9, 75.0,
77.6, 77.3, 73.9, 74.6, 75.5, 74.0,
75.9, 72.9, 73.9, 74.2, 79.1, 75.4,
7 6.3, 7 5.3, 76.2, 7 4.9, 79.0, 7 5.1
a) Calculate a point estimate of the
mean pull-off force of all
connectors in the population State
which estimator you used and why
b) Calculate a point estimate of the
pull-off force value that separates
4.4* Gia sft ta c6 mdu ngSu nhi6n kichthu6c 2n tir t6ng th6 kli hiQu ti X,
E(x)=lr ve V(X)=02.Dit
x,=*It vd X, =;lt*,
li 2 u6c luo ng cria p U'dc lugngnio ld t6t hon? Giii thich
4.5** Gia s0'X1, X2, ,X61i miu nglu
nhi€n tir t6ng th6 v6i trung binh p
vi phuong sai o2 X6t c6c u6c
lu-o-ng sau cria p:
the weakest'50% of the connectors
in tlre population from thestrongest 5002
c) Calculate point estimates of the
population variance and the
population standard deviation
4.7 Xn and Sf are the sample meanand adj sampte variance from apopulation with mean p and
variance
"! Similarly,X2 and Sl ur" the sample meanand sample adj variance from a
second independent populationwith mean p2 and variance ol.
The sample sizes are n,and n2,respectively
a) Show that X, - X2 is anurrbiased estimator of lrt - pz .
b) Find the standard error of
X, - X, How could you estimate
the standard error?
4.8 Let X be a Bernoulli randomvariable The probability massfunc is
r(*;p) = {0 (' -P)'-*' x = o'l
where p is the parameter to beestimated Find the likelihoodfunction and the loglikelihood one
of a randorn sample of size n
c) Tinh u6c lugng di€m cta phuong sai t6ng th6 vn dQ tQch
chuin t6ng thii
.DS a) i=75,60;
b) fD = 74,28;
c) S=1,6967
4.7 Xnva Sfl ta trung binh m6u vd
phuong sai m6u hi€u chinh tirmQt t6ng th.3 vdi trung binh p
vd phuong sai of Tuong tg,X2 va Sl ld trung binh m6u vd
phuong sai m6u higu chinh tir t6ngth6 d$c l4p thri hai v6i trung binh
p2 vd phuon g sai oj Kich thu6cmdu tucrng ring Id n1, n2
lugng kh6ng chQch cria pt - ltz .
b) Tim dO l€ch chu6n 'cfia
X,-X, Bgn u6c luqng d9 lQchchuAn ndy th6 nio?
DS: b)
4.8 Cia sti X ld BNN Bernoulli Hdrnkh6i luqng x6c sudt ld:
(f(x;p) =] n- (r -P)'-*' x = o'l
[0, nguo c laitrong d6 p ld tham sti cAn u6c
lugng Tim hAm hSp ly vi logahim hqp lj cira miu ngiu nhi€nkich thu6c n
Trang 184.9 In the normal distribution case, the
maximum likelihood estimators of
p and o2 were
Find the maximum likelihood
estimator of the function
r''(u,"2)= ,f' = o.
4.10** Transistors have a life that is
exponentially distributed with
parameter 1, A random sample of
' n transistors is taken What is
the
joint probability density function
of the sample?
4.11* ASTM Standard E23 defines
standard test methods for notched
bar impact testing of metallic
materials The Charpy V-notch
(CVN) technique measures impact
energy and is often used to
determine whether or not a
material experiences a
ductile-to-brittle transition with decreasing
temperature Ten measurements of
irnpact energy (J) on specimens of
impact energy (J) on specimens of
4238 steel cut near 60oC are as
Tim udc lugng hqp lli cgc tl4i cfra
D^t
4.10** fu6i thg cria c6c b6ng tldn rtiQn
tri c6 ph6n bii mfi vdi tham sii )'
MQt mdu nglu nhi6n kich thuo c ntlugc rtt ra Tim hdm mflt tlQ <tdng
thdi cfia miu.
pg 1n"_l(x1+ +xn), (xi >0).
4.11* Ky thu6t CM,l tlo nlng lugngndn vi thuong tlugc dirng ili x6cttlnh mQt v{t liQu c6 thay aOi ttrd6o sang ddn hay kh6ng khi nhiQttlQ giim ddn Ph6p do nlng luqngn6n (J) tr6n nhi1ng miu th6p A238
4.12x* A confidence interval estimate
is desired for the gain in a circuit
and the calcium concentration
(milligrams per liter) measured A
90% Cl on the mean calciumconcentration is 0.49 < p < 0.82
a) Would a 99Yo CI calculated
from the same sample data been
longer or shorter?
b) Consider the following
statement: There is a 90Yo chance
that p is between 0.49 and 0.82 Is
this statement correct? Explain
c) Consider the following
, statement: If n = 100 randomsarnples of water from the lakewere taken and the 90% CI on pr
computed, and this process wasrepeated 1000 times, 900 of theCIs will contain the true value of
DS: (64,478+0,653)
''n
4.12**, Ngudi ta mu6n u6c lugngkhoing tin c{y cho clQ suy giim trong mQt m4ch cria thii5t bi ben
d6n Gii st} tls suy giim c6 ph6nbi5 chuAn vbi o :20.
a) Tim khodng tin cQy 90%o cho 1tkhi n=20,f =1000
b) Tim khoing tin cfy 99Yo cho pt
canxitrung binh ld 0,49 < p < 0,82.
a) Khoing tin cfy 99% tfnh tluoc
tt mdu nu6c d6 ld rQng ra hay hgp
c) Xdt ph6t bi6u sau ddy: NCu 6y
ngdu nhi6n 100 miu nu6c tir h6 vdtinh ra khoing tin cfly 90%o cho 1t,
vd qu6 trinh niy lap lai 1000 lAnthi c6 cO 900 khoing tin cgy chfa
p Phrit bi6u ndy c6 dfrng kh6ng?
Giiithich c6u tri loi c[ra ban