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SIN601S - STATISTICAL INFERENCE 2 - 2ND OPP - JAN 2023 |
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nAmlBIA unlVERSITY
OF SCIEn CE Ano TECHn OLOGY
FACULTYOF HEALTH,APPLIEDSCIENCESAND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BSAM
LEVEL: 6
COURSE CODE: SIN601S
COURSE NAME: STATISTICALINFERENCE2
SESSION: JANUARY 2023
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY/ SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
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.
PERMISSIBLEMATERIALS
1. Nonprogrammable scientific calculator
THIS QUESTION PAPERCONSISTSOF 3 PAGES(Including this front page)
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Question 1 [28 Marks]
1.1. Let Y1 < Y2 < ···< Yn be the order statistics of n independently and identically distributed
continuous random variables Xi, X2, ..., Xn with probability density function f and cumulative
distribution function F. Then, the cumulative distribution function of r th order statistics, Fyr(y)
is given by
=LC) n
Fyr(y)
(Fx(Y))k (1- Fx(Y))n-k
k=r
Use this result to show that the cumulative distribution of the minimum statistic is given by
[4]
1.2. Let Y1 < Y2 < ... < Y5 be the order statistics of 5 independently and identically distributed
continuous random variables X1, X2 , ... , X5 with pdf f given by
fx(x) = {6x2 for O x < 1
0 otherwise
Then
1.2.1. Show that the cumulative density function of Xis, Fx(x) = 2x 3
[2]
1.2.2. find the pdf of the r th order statistics
[3]
1.2.3. find the pdf of the minimum order statistics
[3]
1.2.4. find the pdf of the maximum order statistics
[3]
1.2.5. find the pdf of the median
[4]
1.2.6. find the joint pdf of the l't and 5th order statistics
[5]
Hint: fr;. riYi, Yj) = (i-l)!U-~~l)!(n-j)! [F(yi)ji-l f(yJ[F(yj) - F(yJ ti-i f(yj) [1 - F(yj )r-j
fyr(y) = (r-1)7~n-r)/x(y)[Fx(y)V-1[1 - Fx(y)]n-r
1.3. Let Y1 < Y2 < ···< Yn be the order statistics of n independently and identically distributed
continuous random variables X1, X2 , ... , Xn with standard normal, N(O, 1), then find the joint pdf
Yi, Y2, ..• , Yn-
[4]
Question2 [11 Marks]
2.1. Let Xi, X2 , .... , Xn be independently and identically distributed random variable with normal
= = distribution having E(Xa µ and V(Xi) (J 2 . Then show, using the moment generating function,
that Y = IP=iXi has a normal distribution with mean µy = nµ and variance o} = ncr2 . (Hint: If
cr2c2
= X~N(µ, (J 2 ), then Mx(t) eµt+-2-).
[8]
2.2. Let Xi, X2 , ... , Xn be a random sample from a normal distribution with meanµ and variance (J 2 .
= ~x n
-2
2
Then find the variance of S 2
Li=iCXi-x) . Hint: (n -1)
n-1
\\u
2 (n - 1) with mean (n -1) and
variance 2(n - 1)
[3]
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Question 3 [30 Marks]
3.1. The length of life of a component operating in guidance control system for missiles is assumed to
follow a Weibull distribution with density function
A(;) T k X· k-1 (x-)k
f(xd.:l) =
e- ,xi~ 0
{ 0,
elsewhere.
If the parameter k is assumed to known, then find the MLE of X
[10]
3.2. Let Xi, X2 , ... , Xn be a random sample from a normal population with mean µ and variance a-2 •
That is
f(xdµ,
a-2 ) = r-1c
l(Xi-µ2)
e-2 (l
for
- oo < x < oo; -oo
< µ < oo and a-2 > 0
v2rra
3.2.1.
3.2.2.
3.2.3.
What are the method of moment estimators of the meanµ and variance a-2 ?
[9]
Lf= Ifµ is known, then show that 1 (xi - µ) 2 is sufficient statistic for a-2 .
[5]
If a-2 assumed to be known, derive the 100(1 - a)% Cl for µ using the pivotal quantity
method.
[6]
Question 4 [21 Marks]
4. Let X1,X2,..., Xnbe and independent Bernoulli random variables with probability of successp and
probability mass function
f(xdp) = {f x·c1 -p )1-x·' for xi= 0, l
otherwise
4.1. Using the mgf of X, show that the mean and variance of Xi are p and p(l - p), respectively.
(Hint: Mx(t) =pet+ (l - p)}.
[5]
4.2. Show that the Xis a minimum variance unbiased estimator (MVUE) of p.
[16]
Question 5 [10 Marks]
5. Suppose the prior distribution of 0 is uniform over the interval (2, 5) with pdf given by
h( 0) = (~ if 2 < 0 < 5
0 otherwise
Given 0, Xis uniform over the interval (0, 0) with pdf given by
f(xl0) = (¼o if o < x < 0
otherwise
What is the Bayes' estimate of 0 for an absolute difference error loss if the sample consists of one
observation X =1?
[10]
=== END OF PAPER===
TOTAL MARKS:100
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