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SIN601S - STATISTICAL INFERENCE 2 - 1ST OPP - NOV 2022 |
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nAmlBIA unlVERSITY
OF SCIEnCE Ano TECHn OLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND 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: NOVEMBER 2022
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 calculator
THIS QUESTION PAPERCONSISTSOF 3 PAGES{Including this front page)
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Question1 [29 Marks]
1.1. Let Y1 < Y2 < ... < Yn be the order statistic of n independently and identically distributed
continuous random variables X1, X2, ... , Xn with probability density function f and cumulative
distribution function F. Then, the cumulative distribution function of r th order statistics, Fv/Y) is
given by
LC) n
Fy/y) =
(Fx(Y)l (1 - Fx(Y)r-k
k=r
Use this result to show that the marginal distribution of the rth order statistic is given by
(10]
1.2. Suppose the random variables X1 , X2 , ... , Xn are independently and identically distributed
exponentially with the parameter 0, that is
f(x) = {0e-0x, X > 0
0,
elsewhere
Let Y1 < Y2 < ... < Yn be the order statistics for X1, X2 , ... , Xn. Then,
= 1.2.1. Show that the cumulative density function of Xis, Fx(x) l - e- 0x
[3]
1.2.2. Find the probability density function of the minimum order statistic Y1
(4]
1.2.3. Which density function does the p.d.f of Y1 belongs to?
[1]
1.2.4. Find the joint p.d.f. of Y1 , Y2 , ... , Yn
(4]
1.2.5. If n = 5 and 0 = 0.5, then find
1.2.5.1. the probability that the sample maximum is greater than 2.
(4]
1.2.5.2. the probability density function of the median.
[3]
Question2 [12 Marks]
2.1. Let X1, X2 , .... , Xn be independently and identically distributed random variable with normal
= = distribution having E(Xi) µ and V(XJ cr2 . Then show, using the moment generating
x-/:: (µ, function, that Z =
has a standard normal distribution. (Hint:If X~N
2
a ),
ahn
n
= then Mx(t)
cr2t2
eµt+""'zn ).
(9]
2.2. Let Xi, X2 , ... , Xn be a random sample from a normal distribution with meanµ and variance cr2 .
= Then find the expected value of S 2 L<n=1n(-X1·' -X)2 .
Hint: (n -1) sa: ~x2 (n - 1) with mean (n - 1) and variance 2(n - 1)
[3]
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Question 3 [23 Marks]
= 3.1. A random sample of n observations Xi, X2, ... , Xn is selected for a population Xi, for i
1, 2, ... , n which possessesa gamma probability density function with parameters a and 0. Use
the method of moment to estimate a and 0.
= (Hint: If X~Gamma(a, 0), then the Mx(t) (1 - 0t)-a)
[10]
3.2. Let X1 , Xz, ... , Xn denote a random sample from a distribution with density function
fx(xl 0) = {Cl- 0)x- 0 for O<_x < l
0
otherwise.
Find maximum likelihood estimators of 0.
[7]
= 3.3. Observations Y1 , ... , }; 1 are assumed to come from a model with E(Ya 2 + 0 xi where 0
is an unknown parameter and x1 , x 2, ... , Xn are given constants. What is the least square
estimate of the parameter 0?
[6]
Question 4 [26 Marks]
4.1. Let X1, X2 , ... , Xn denote a random sample a Rayleigh distribution with parameter 0.
= fx(xl 0) {20xe- 0x 2 for x >_0 and 0 > 0
.
0
otherwise
Show that I:f=1 x'fis sufficient for 0
[5]
4.2. Suppose a random sample X1, X2 , ... Xn is selected from a normally distributed population with
unknown meanµ and variance (5 2 .
4.2.1. Show that Xis a minimum variance unbiased estimator (MVUE) ofµ.
[15]
4.2.2. Derive the 100(1 - a)% Cl forµ using the pivotal quantity method.
[6]
Question 5 [10 Marks]
5. Suppose one observation was taken of a random variable X which yielded the value 2. The density
function for X is
!¾ f(xl 0) = for O < x < 0
0 otherwise
and the prior distribution of 0 is
{20- 1 h( 0) =
2 for
0 < oo
0 otherwise
5.1. Find the posterior distribution of 0.
[7]
5.2. If the squared error loss function is used, find the Bayes' estimate of 0.
[3]
=== ENDOF PAPER===
TOTALMARKS:100
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