MST 561 - Statistical Inference - EPrints USM

UNIVERSITI SAINS MALAYSIA First Semester Examination Academic Session

200812009

November 2008

MST 561 - Statistical Inference I

^Pe

ntaab

i

ran Stati sti k]

Duration

:

3 hours [Masa

^{: 3}

jam]

Please check that this examination paper consists of TWELVE pages of

^printed

materials before you begin the examination.

[Sila pastikan bahawa kertas peperiksaan ini mengandungi DIJA BELAS muka surat yang bercetak sebelum anda memulakan peperiksaan ini.l

Instructions : Answer all five [5] questions.

tArahan : Jawab semua lima [5] soalan.l

...2t-

IMST 561I

Three employees,

A,

B and C in a

firm

have been identified.

A manager and

an assistant manager

will

be appointed among these three employees.

(i)

^What is the set of all possible outcomes,

f)?

(ii)

^{What is the}

o-field,

^{S, of this} experiment based on (i)?

[30 marks]

A point is chosen randomly from the interval (0,1) and let the

random variable

X,

represent the number which coruesponds to that point.

Then

^choose

a point at random from the

interval (0,", ),

^where

x,

is

the

experimental value

of Xr;

^and let the random variable

X,

represent

the

^number

which

corresponds to this point.

(D

^{Find the} marginal probability density function

of X,

^.

(ii)

Find the conditional probability density function

of Xr, given Xr

_{= xr.}

(iii)

^{Compute P}

(X, + Xr>l)

^.

[40 marks]

Let p(x) =l /rY * | ,x:1,2,3, ...,zeto

elsewhere, be the

probability

mass function

\/)

of the random variableX.

(i)

^Find the moment generating function

ofX.

(iD

^Find the mean and variance

ofXusing

(i).

[30 marks]

2

t. (a)

(b)

(c)

IMST

⁵⁶¹¹

Tiga

kakitangan

A, B dan C dalam

suatu

firma telah dikenalpasti.

Seorang

pengurus dan seorang penolong pengunts akan dilantik di kalangan

tiga kakitangan ini.

(i)

Apakah set semua kesudahan yang mungkin, (27

(ii)

Apakah

medan-q

S, untuk eksperimen

ini

berdasarkan (i)?

[30

^markahJ

Satu titik dipilih secara rawak dari selang (0,1) dan biarkan

pembolehubah

rawak X, mewakili

nombor yang _sepadan dengan

titik itu.

Kemudian

pilih

satu

titik

secara rav,ak

dari

selang

(0,r,),

yqng mana

x, ialah nilai ujikaji bagi x,;

dan

biarkan pembolehubah

rawak x, mewakili

nombor

yang

sepadan dengan

titik

ini.

(i)

Cari

fungsi

ketumpatan kebarangkalian sut

bagi

_{X ,.}

(ii)

Cari.fungsi ketumparan kebarangknlian bersyarat X

,, diberi

^X

t

^{= xt.}

(iiil Kira P(X,+ X, > I)

^.

[40

^markahJ

/ rY

Biarkan p(x)=[ ;l

^,

x: ], 2, 3, ..., sifar di

tempat

lain,

sebagaifungsi

jisim

\2.,

keb ar angkalian b agi p e mbol ehub ah raw ak X.

(i) Carifungsi

penjana momen bagi X.

(ii) Cari

min dan varians bagi

X

dengan menggunakan (i).

[30

^markahJ

3

I (a)

f.

(b)

(c)

...41-

4

(a)

Consider the random valiable

Xwith

probability mass function

f

,'l

P(X :

x)

: p,

₌ lYl

;,

^x

:

^{0, 1,2,}

...,

^n.

Find the

probability

generating function

ofX.

[Hint : (l+r)'

₌

tf 1)r't

=\J

^I

IMST

^561I

2.

(b)

(c)

[30 marks]

Let

T

= Y, , where W

^atd,

V

are respectively,

the

standard

normal

random

^lvl''

variable and the chi-square random variable

with r

degrees

of

^freedom.

By

using

the

necessary

theorems, show that T2 has an F distribution. What is

the distribution

of

I_^ ^?

T"

[30 marks]

Let X,

^and

X,

have the

joint probability

mass

function p(x,,xr)=*, xt = l,

2,

3

and xz

: I,2,

3, zeto elsewhere.

(i)

Find the

joint

probability mass function

of {

⁼

X,X,

^{and Y, = X z.}

(ii)

Find the marginal probability mass function

of {

^.

[40 marks]

IMST 561I

.1. Pertimbongkan pembolehubah rawak

X

dengan fungsi

jisim

kebarangkalian

P(X

₌

x)= p, =V,x : t:)

0, 1,2, ..., n.

Cori fungsi penjana kebarangkalian ^{bagi X.}

[Pettn : (t

⁺

D' =Z\]),' l

[30

^markahJ

Biarkan

T ₌

-j

W

, yang

^mana

I( dan

^{V adaloh} masing-masing pembolehubah

rlv l,

rawak

normal

piawai

dan pembolehubah

rawak

khi-kuasadua dengan

r

darjah kebebasan.

Dengan

menggunakan

teorem tertentu, tunjukkan bahawa

^T2

mempunyai taburan

F.

Apakah taburan untuk

L

^?

[30

^markahJ

Biarkan x t dan x 2

^mempunyai

fungsi jisim kebarangkalian

tercantum

/ \

X,X.

plx,,rr)=if , xt : I,

_{2, 3}

dan

_xz

: I,

_{2, 3,}

sifar di

tempat lain.

(i)

Cari fungsi

jisim

kebarangkalian tercantum

bagi

_{Y, = X ,X}

, dan

^Y,

-

^{X 2.}

(ii) Carifungsi jisim

kebarangkalian sut

bagi

Y,.

[40

^markahJ

...6/-

(a) 3.

(b)

lMSr 561I Let Y,

denote

the minirnum statistic of a random

sample

of size n from

a

distribution

that has

probability

density

function f(x)=u-G-g), 0 < x <

^{co, zero}

elsewhere.

Let 2,,

₌ ,(Y,

- e).

^{nina the}

limiting

distribution

of Z,

^.

[30 marks]

Let Xr,X2,...,Xn be a random

sample

from a Bernoulli, b(l,e)

distribution, where

0 < e < l.

^Then

, =f*, follows

^a

binomial, Bin(n,0)

distribution

with

i=l

probability

mass function

f(

:'\''(1 - s;'-r, v

=

0,r,2,"',n

f(yl})={[Y/" '^ )

^'r

Assume that rhe

*1", *"u]blt,, d."'rl',;;':ion orthe

random variable @ is given by

Ir("

^{+ P)}

^*

r(e)= ]ffie"-t(1 -e)o-''

o < o < I

[. 0,

^elsewhere

where

a

and _{B are} known positive constants.

(i)

Find the posterior probability _density function

for

@.

(ii) Find

the Bayes' solution,

w(y) for 0 with

respect

to

the

prior

probability density function, ft(O) using the loss

function, tle,w1y1l=

_[e

- .U)]'

^.

[40 marks]

(c) Let X,,X2,...,X

_{n be} a random sample

from

a beta distribution

with

the

following probability

density function:

f (*,;o):

fi#t',(t -',)l'-' ,

⁰

", < 1,

^e

>

^o

Find a sufficient statistic for 0.

[30 marks]

18"'

[qotltow

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'e !30q dn4nc 4ltsltzls ^Wnns

rnJ

o <e'[ >'x > o',-,f('* - n'4#H=(s:,x)!

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p

dwn1 a4 p3 unl un& u a

p Dnq uolnqq npndlnp yul^oJ pdwns nwns

mSnqas

uX,..-,rX,,X

_uzyJorg

[qn4nw

6pJ

'

l{,0^-

^ef

=l(,r),n'e]t

un13nta4 p8unt

um1ounSSuaw

un8uap (ilLl '1tot,td unrp4Sunnqay

unpdrun1a4

63unl

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^'

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Lt'"''e '['0= t 'n-,(0- l-e(,\l=(elt)t y, ))

untlo4Suntoqay _rys1 { ls8un! un&uap (g'u)u1g

l=l

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'tilnoung uolnqol opodltop ,o^r, pdruos

rlrryovaut

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[qotltow

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^@

> x > e @_,)_o=(x)l uor1ry8uonqat1 unpdwnla4 ls8unl mtundutaw Suot uornqu opodltop

u zrDS un8uap )qMDJ _Tadtuns

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4.

IMST

⁵⁶¹¹

(a) Let X,,X2,...,X,, denote a random sample of size n > 2 from the

N(O,e) distribution.

(i)

Find the maximum likelihood estirnator of 0.

(ii)

Find the Cramer-Rao lower bound

for

the variance of unbiased estimatols

of0.

(iii)

Is the maximum likelihood estimator of 0 in

(i)

an

efficient

estimator of 0?

[40 marks]

(b)

Assume that

X,,X2,...,X,

^is a random sample having an exponential probability density

function,

_-f ^(x;?,) = 7"e-b, x

)

0. Show

that i

=

I,

is not consistent for ^1,.

[20 marks]

(c)

Assume

thatXis

a single observation from a distribution

with

density function

(i)

(iD

(iii)

"f(*;u)

⁼

ae-*,

⁰

<x

^{< oo, ct}

)

0.

Is2aX apivotal

quantity? Explain.

Find the

confidence

coefficient for the random interval (X, 3n if

^this

interval is ^a confidence interval

for

cr.

What is the mathematical expectation

of

the length

of

the random interval

in (ii)?

[40 marks]

...9/-

/

(a)

+.

e [MSr

^551]

Biarkan Xt,X2,...,Xn menandai suatu sampel rawak dengan saiz n >

²

daripada taburan N(0, 0).

(,

Cari penganggar kebolehjadian maluintum bagi

(ii) Cari batas bawah

Cranter-Rao

untuk

varians salcsama 0.

(ii,

Adakah

penganggar

kebolehjadian malrsimum penganggar cekap

bagi il

[40

^markahJ

Andaikan bahawa X

t,X

2,...,X

,

^{suatu sampel}

rawak

^dengon

fungsi

^ketumpatan

kebarangkalian

elrsponen,

f(x;A)=)"e-b , x ) 0. Tunjukkan bahawa i=Y,

tidak konsisten untuk ),.

[20

^markahJ

Andaikan _bahawa

X

ialah suatu cerapan tunggal daripada taburan denganfungsi ketumpatan

0.

p e ngqngg ar -p e ngan ggar

bagi 0 dalam (i)

suatu

(b)

(c)

(,

(ii) (iii)

"f(x;a)= d,€-*,

0

1 x < o, d>

^0.

Adakah 2aX suatu kuantiti pangsian? Jelaskan.

Cari pekali

keyakinan

bagi

selang

rawak (X,

3X)

jika selang ini

_ialah

selang

keyakinanbagi a

Apakah

jangkaan

matematik bagi panjang selang rawak dalam

(ii)?

[40

^markahJ

...1U-

lMSr

⁵⁶¹¹

5. (a) Let X,,X2,...,Xrc

denote a random sample

of

size 10

from

a Poisson distribution

with

mean

0. Show that the critical region C defined U, it, ) c' is a

^best

t=l

critical

region

for

testing 110

:0

= 0.1 versus

Hr:0:0.5. If

c'

:3,

_determine

for

this test, the size of the Type-I emor,

a

^and the power at 0

:

_0.5.

[40 marks]

(b) Let X.,X,,...,X,, denote a random sample having the probability

density

function, f (x;g)

⁼

0e-0',

^x

)

0,

@: {0

: 0

> 0}.

Find the generalized likelihood ratio test

of

size-cr to test H

, :0 (

_0o

vs. 1/,

_{: 0 > 0o .}

[30 marks]

(c)

Assume that

X

is a single observation

from

a

distribution with

probability density function

f@;il=0.trt-' , o1x<1; o>o.

To

test

Ho:0 <1 vs. Hr:0 >1,

the test

that is

used

is reject H0 if

and

only if

X ,+.

^{Find the} power function and the size of this test.

z

[30 marks]

10

lMsr 561I

(a) Biarkan

X

t,X 2,...,X,,,

^{menandai suatu sampel}

rawak

saiz

l0 daripada

taburan

Poisson

dengan

min 0.

Tunjukkan bahawa

rantau genting C yang

ditalcriJkan

10

oleh lx,) c'

adalah sttatu rantau genting

terbaik

tmtuk menguji

H,,:0

_=0.1

lawan H,:0=0.5. Jika c' : 3, cari

_tmtuk

ujian ini, saiz ralat

Jenis-L,

a

dan kuasanya

pada 0:

^0.5.

[40

^markahJ

(b) Biarkan X.,X2,...,X,

menandqi

suatu

sampel

rawak yang

mempunyai

fungsi

ketumpatan kebarangkalian,

f(x;O)

^{= 0}

e-t*, x ) 0,

@

: {0 ;

_e

> 0}. Cari

ujian nisbah kebolehjadian

teritlak

saiz-a untuk

menguji H,, :0

1 0,,

lawan

H

, :0

^{> 0,, .}

[3A markahJ

(c) Andaikan

bahawa

X adalah suatu

cerapan

tunggal daripada taburan

dengan fun ^gs

i

^{ke tump at} an ^{ke b} ar an gkal i an

f (x;0):?xe-''

0

<x < 1; e>

^0'

11

J.

Untuk

menguji H,,:011 lawan H,:0>1, ujian yang

digunakan adalah tolak H,,

jika

dan hanya

jika X

>

!

_{, Corf}

frngsi

kuasa dan saiz

ujian

ini.

[30

^markahJ

...12t-

12

APPBNDIX

I

LAMPIRAN

IMST 561I

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