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SIN601S - STATISTICAL INFERENCE 2 - 1ST OPP - NOVEMBER 2023 |
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nAmlBIA UnlVERSITY
OF SCIEnCEAnDTECHnOLOGY
Facultyof Health,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
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 CODE: SIN601S
COURSE NAME: STATISTICALINFERENCE2
SESSION: NOVEMBER 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Dr D. B. GEMECHU
MODERATOR:
Dr D. NTIRAMPEBA
INSTRUCTIONS
1. There are 5 questions, answer ALL the questions by showing all the necessary steps.
2. Write clearly and neatly.
3. Number the answers clearly.
4. Round your answers to at least four decimal places, if applicable.
PERMISSIBLE MATERIALS
1. Nonprogrammable scientific calculators with no cover.
THIS QUESTION PAPERCONSISTSOF 3 PAGES(including this front page) and 4 EXTRA
x ATTACHED STATISTICAL TABLES {Z-, t-, 2- and F-distribution tables)
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Question 1 (20 marks]
1. Let Y1 < Y2 < ···< Yu be the order statistics of 11 independently and identically distributed
continuous random variables X1 , X2 , ... , X11 with pdf f given by
Then find
fx(x)=}~, forO<x<3
lo, otherwise
= I.I. The pdf of the r th order statistics. Hint: fy/y)
(n-r)~~r-l)!
[Fx(y)y- 1 [1- Fx(y)]n-rfx(Y)
(3]
1.2. The pdfofthe minimum order statistics
13)
1.3. The pdf of the maximum order statistics
[3)
1.4.The pdf of the median
[3]
1.5.The joint pdf ofYv Y2 , ... , Yu
[31
1.6. Ifthe number ofrandom variables is reduced to 3, thus, X1 ,X 2 , X3 , then find the joint pdf of the
minimum and maximum order statistics.
[41
= Hint: fY1,y/y,, Yj) (i-l)!U-~~l)!(n-j)! [F(yJ]i-1 f (yi) [F(yj) - F(yi) f-i-1 f(Yj) [1 - F(yj)r-j
Question 2 (22 marks]
2.1. If the random variables Xi, ... , Xm are independent and if Xi has the x2 distribution with k
= degrees of freedom (i 1, ... , m), then show, using the moment generating function, that
the sum Y = X1 + . . . + Xm has the x2 distribution with km degrees of freedom. Hint:
k
use
Mx (t)
l
= (-1-21-t)
2.
[81
2.2. If z1, ... , z20 denote a random sample from a standard normal distribution, find the value of c
such that P(.Et1~zf $ c) = 0.25
[41
= = 2.3. Consider two independent samples of size n 1 8 and n 2 12 from two normal populations
with populations variances c,f= 3c,fT.hen, find the value b of such that Pe: ::;b)= 0.95.
(51
2.4. Let the random sample X1 ,X2 , ... , Xn ~N(µ, c, 2 ) where bothµ and c, 2 are unknown. Derive the
100(1 - a)% Cl forµ using the pivotal quantity method.
(51
Question 3 [25 marks]
3.1. Let X1 , X 2, •.. , Xn denote a random sample a Rayleigh distribution with parameter 0.
fx(xiJ0) = f20xie- 0xt2, for xi~ 0 and e > 0
lo,
otherwise
3.1.1. Find the maximum likelihood estimator of 0
(6)
3.1.2. Find the maximum likelihood estimator of g(0) = -le+ 2
(3)
3.1.3. Show that If=ixf is sufficient fore
(4)
3.2. Let 0 > 0 and let X 1 , ... , Xn be a random sample of size n from a distribution of pdf
f(x;
0)
=3
03
x
2
,
for
O< x < 0
Find the estimator of 0 using the method of moments.
(6)
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3.3. Let Yi, Y2, .•• , Yn be n independent random variables such that each ~~Poisson({Jxi), where {J
is an unknown parameter. If {(yi, x 1), (y 2, x2), ... , (yn, Xn)} is a dataset where Yi, y2, ... , Yn are
the observed values based on x 1, x2, ... , Xn, then find the least estimator of fJ.
[6]
Question 4 [12 marks)
4. Let X1 , ... , Xn be Poisson random variables with parameter f3with probability massfunction
pxe-/3
f(xl{J) =-,-fxo.r
x = 0,1,2, ...
= Assume that f3has an exponential prior distribution with 0 1 and probability
density
function
h(/3) = fe-/3, (3 > 0
lo, elsewhere.
4.1.Showthatthe posterior distribution of fJis a f(If=1 xi + l,n + 1). Hint: If X~f(a,{3),
/3 a-1 -{3x > 0 f3 0
fx(xla,{3) = {rca)x 0, e
' x - ,a, >
elsewhere
with E(X) =
then
(8)
4.2.Compute the posterior mean of~
(4)
QUESTION 5 [21 marks)
5. If X1,X 2, . .• , Xn be a random sample from the Poisson distribution with the parameter 0, then
= 5.1. Show that the mean and variance of Xi are 0. Hint: MxJt) e 8(et- 1)
(6)
5.2. Show that Xis a minimum variance unbiased estimator (MVUE)of 0.
[12]
5.3. Show that Xis also a consistent estimator of 0.
[3]
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Table for a=.05
J\\
Idf2/dfl
1
2
3
4
5
6
7
8
9
11
F (.0S,df1,df2)
I
2
3
4
s I6
7
8
9
10
12
I I I I I I 161.448 199.soo 215.101 224.583 230.162 233.9861 236.7681 238.8831 240.543 241.882 1 243.906
I 18.5131 19.ooo 1 19.164 19.2471 19.2961.---1-9_.3_2·,_-19--1 9_.3_5_3-,,-1.--937_1_1,--1_9_.3_841-9.3961 19.413
1.------,-----.---9.-27_7_,....
10.128 1 9.5521
I 7.7091
6.9441
6.591
__ _
9.111 1
I 6.388
9.0141,---8-.9-4-1·.----8-.8-87-1
6.2561 6.163
6.09421
8.845 I 8.812
6.041 ,.--5-.9-98-
8.786 1 8.745
I 5.964 5.912
6.6081 5.7861 5.409
5.9871
I 5.591
I 5.318
5.143 1
4.7371
4.4591
4.757
4.347
4.066
5.1921
4.533
4.120
3.838
I 5.050 4.950
4.3871 4.284
3.972 3.866
3.688 3.581
4.8761
4.2071
3.7871
3.501 1
4.818 I 4.772
4.147 ,-i -4-.0-99-
I 3.726
3.676 ,...I
---
4.735
I 4.060
3.6371
3.4381 3.3881.- ---3.347
4.6781
3.999
3.5751
3.284
5.1171 4.2561 3.863 3.633 3.482 3.374
I 3.293 I 3.2291 3.178 3.137
3.073
4.9651
4.8441
I 4.103 3.708
3.9821 3.587
3.478
3.358
3.326
3.204
3.217
3.095
3.1361
I 3.012
I 3.072
2.9481
I 3.020
2.8961
2.978
2.854
2.913
2.788
12
4.7471 3.8851 3.490 3.259 3.106 2.996
2.913 1 2.8491 2.7961 2.753 2.687
13
4.667
14
4.600
15
4.543
16
4494
17 I 4.451
I 18
4.414
I 19
4.381
I 20
4.351
3.8061 3.411 3.1791 3.025 2.915
3.739 1 3.344 3.112 2.958
., ., I -· - 3.682 ,-1-3-.2-87-.-,-- 3.056
2.901
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3 001 .----?-8-5?-
3.591 1 3.1971 2.965 2.810
2.848
2.791
2 741
2.699 1
3.5551
I 3.522
I 3.493
3.160 1
3.1271
3.0981
2.928
2.895
2.866
2.113 1
I 2.740
2.111 I
2.661 1
2.6281
2.5991
2.8321
2.764
2.707
2 657
2.614
2.577
2.544
2.514
2.7671
2.699
2.641
2 591
2.548
2.510
2.477
2.441
2.714
2.645
2.671 2.604
i 2.602 2.534
2.587 2.544 2.475
?-·537
2.494
2 494
2.450
?-·425
2.381
2.456 2.412 2.342
2.4231 2.378
I 2.393 2.348
2.308
2.278