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SIN601S - STATISTICAL INFERENCE 2 - 1ST -NOVEMBER 2024 |
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nAml BIA un IVERSITY
' OF SCIEnCE
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
Statisticsand ActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264 612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM
LEVEL: 6
COURSE:STATISTICAL INFERENCE 2
COURSECODE: SIN601S
DATE: NOVEMBER 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
EXAMINER:
MODERATOR:
FIRST OPPORTUNITY EAMINATION QUESTION PAPER
Dr J MWANYEKANGE
Dr D. B. GEMECHU
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Write your answers neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator without cover
This paper consists of 4 pages including this front page
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Question 1 [15 Marks]
1.1. Let Y1 < Y2 < ···< Y5 be the ordered statistics of 5 independently and identically
distributed continuous random variables Xi, X2, .•• , X5 with pdf f given by
{6x2, fx(x) =
for O < x < l
0, Otherwise
Then,
1.1.1. Show that the cumulative density function of Xis Fx(x) = 2x 3
[3]
1.1.2. Find the pdf of the minimum order statistics
[3]
1.1.3. Find the pdf of the maximum order statistics
[3]
1.1.4. Find the joint pdf of the 2nd and 5th order statistics
[6]
= Hint: fv;,v/Yi,Yj)
(i-l)!(j-~~l)!(n-j)!
Fx(Yj)]n-j fx(Yi)fx(Yj)
[Fx(Ya]i- 1 [Fx(YJ - Fx(Yaf-i-l[ l -
Question 2 [13 marks]
2.1 Let Xi, X2, .•. , Xn be independently and identically distributed random variable with normal
distribution having E(Xi) = µ and Var(Xi) = a 2 •
Show, using the moment generating function, that Y = 2:~1 Xi has a normal distribution
a; with µy = nµ and = na 2•
[8]
(Hint:
If X~N(µ, cr2 ), then MxJt)
=
a-2t2)
eµt+-2-
2.2 If X ~X~ and Y~x~ such that X and Y are independent, what is the distribution of X + Y? [5]
Question 3 [8 marks]
3. Let Xi, X2, ... , Xn be a random sample of observations from a Bernoulli distribution
Show that T =
is an unbiased estimator of 0 2 where y = 2:f=1 xi. (Hint: E(y) = n0
and Var(y) = n0(l - 0)
[8]
Statistical Inference 2 {SIN6015)
l51 Opportunity November 2024
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Question 4 [44 marks]
4.1 If X ~f(a, 0) a random sample ofn observations Xi,X 2, .•. , Xn is selected from a population
= Xi for i 1,2 ... , n posses a gamma probability density function with parameters a and 0.
Use the method of moment to estimate a and 0. Hint: E(X) = a0, Var(X) = a0 2 and
[8]
4.2 Let Xi, X2 , ... , Xn be a random variable from a Bernoulli distribution with pdf:
f(X;p) = px(l-p)1-x,
X = 0,1.
4.2.1 Using the m.g.f of X, show that the mean and variance of Xi are p and p(l - p),
respectively (Hint: MxJt) =pet+ q)
[6]
4.2.2 Find the maximum likelihood estimate of p.
[6]
rr=l 4.2.3 Let Tn =
xi, show that Tn is sufficient for p.
[8]
4.2.4 Show that the X = n If-- 1 Xi is a minimum variance unbiased estimator (MVUE)
of p. (Hint: CRB =
1
a
2)
[16]
nE(aplogf(x;p))
Question 5 [20 marks]
5.1 Suppose that the prior distribution of 0 follow a Gamma distribution with shape a = 2 and
rate {3,
Given 0, X is uniform over the interval (0, 0) with pdf given by
!¾, f(xl0) =
0<x<0
0, Otherwise
(H. What is the posterior distribution of 0.
lnt·
.
m (SIX )
't'
= -fx-"'rr-°((-xxl'l0-0)-)hh-(('BB--))c-t-e )
[8]
5.2 Let Xi, ... , Xn be random samples from the binomial distribution:
Statistical Inference 2 (SIN6015)
1st Opportunity November 2024
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The prior distribution of 8 is a beta distribution with the parameters a and fl,
r(a+p) ea-l (1 - e)P-l for O < 8 < 1
= h(B)
r(a). r(P)
'
.
{ 0,
Otherwise
Show that posterior distribution of 8 given X = x is a beta distribution with parameter x + a
and n - x + {3.(Hint: f.1oex+a-l (1 - e)n+P- 1 de = r(x+a)f(n-x+P))
[12]
f(n+a+l3)
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AL MARKS 100==========================
Statistical Inference 2 (SIN601S)
l51 Opportunity November 2024
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