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PBT602S - PROBABILITY THEORY 2 - 1ST OPP - JUNE 2022 |
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ry
NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of science in Applied Mathematics and Statistics
QUALIFICATION CODE: 07BAMS
LEVEL: 6
COURSE CODE: PBT602S
COURSE NAME: Probability Theory 2
SESSION: JUNE 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Dr D. B. GEMECHU
MODERATOR:
Prof R. KUMAR
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.
Round your answers to at least four decimal places, if
applicable.
PERMISSIBLE MATERIALS
1. Nonprogrammable scientific calculator
THIS QUESTION PAPER CONSISTS OF 4 PAGES (Including this front page)
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Question 1 [12 marks]
1.1. Define the following terms:
1.1.1. Probability function
[3]
1.1.2. Power set
[1]
1.1.3. o-algebra
[2]
1.1.4. Consider an experiment of rolling a die with six faces once.
1.1.4.1. Show that the set a(X) = {9,S, {1,2,4}, {3,5,6}} is a sigma algebra, where S
represents the sample space for a random experiment of rolling a die with six
faces.
[3]
1.1.4.2. Givenaset Y = {(1,2,3.5), {4}, {6}}, then generate the smallest sigma algebra, a(Y)
that contains a set Y.
[3]
Question 2 [24 marks]
2A, Let X be a continuous random variable with p.d.f. given by
x+1, for-1<x<0O,
f(x) {ins for0<sx<1l1,
0,
otherwise.
Then find cumulative density function of X
[7]
2:2: Suppose that the joint CDF of a two dimensional continuous random variable is given by
F. x; = 1—e-*-—eY¥+e"F*), ? ifx>0; ’ y>0 ’
xv (%y) 0,
otherwise.
Then find the joint p.d.f. of X and Y.
[4]
2.3. Consider the following joint pdf of X and Y.
[7]
_ (2, x>0; y>0; x+y<1,
fay) = {0 elsewhere.
2.3.1. Find the marginal probability density function of f(y)
[2]
2.3.2. Find the conditional probability density function ofX given Y = y, fy(x|lY = y)
[2]
2.3.3. Fi, nd P(x <>|1], Y =-1 )
[3]
2.4. Let Y;,¥2, and Y3 be three random variables with E(Y,) = 2, E(¥2) = 3, E(¥3) = 2, of, = 2,
OF, = 3; OF, = 1, OY, Y, = —0.6, Oy, Y, = 0.3, and OY, y, = 2.
2.4.1. Find the expected value and variance of U = 2Y, — 3Y, + Y3
[2]
2.4.2. IfW = Y, + 2Y3, find the covariance between U and W
[4]
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QUESTION 3 [27 marks]
3.1. Let X bea discrete random variable with a probability mass function P(x), then show that the
moment generating function of X is a function of all the moments 4; about the origin which
is given by
fai
t,
t2 ,
tk ,
My(t) = E(e JS lt eit ypbe te tee to
Hint: use Taylor's series expansion: e* = re, = i
[5]
3.2. Let X1,X2,...,X, be a random sample from a Gamma distribution with parameters @ and @, that
IS
Fmla,0) 1=tFepae* gs, 8 -%% fo>r0;ma>0; 6>0,
0
otherwise.
a
3.2.1. Show that the moment generating function of X; is given by Mx, (t) = (=)
[6]
3.2.2. Find the mean of X using the moment generating function of X.
[4]
3.3. Suppose that X is a random variable having a binomial distribution with the parameters n and p
(i.e., X~Bin(n, p))
3.3.1. Find the cumulant generating function of X and find the first cumulant.
Hint: My(t) = (1-p(1—e*))”
[4]
3.3.2. If we define another random variable Y = aX + b, then derive the moment
generating function of Y, where a and b be any constant numbers.
[3]
3.4. Let X and Y be two continuous random variables with f(x) and g(y) bea pdf ofX and Y,
respectively, then show that E[g(y)] = E[E[g(y)X]].
[5]
Question 4 [20 marks]
4.1. Suppose that X and Y are two independent random variables following a chi-square distribution
with m and n degrees of freedom, respectively. Use the moment generating function to show
m
thatX + Y~y2(m +n). (Hint: My(t) = (= y?).
(7]
4.2. If X~Poisson (A), find E(X) and Var(X) using the characteristic function of X.
4.2.1. Show that the characteristic function of X is given by by (t) = er(e"-D
[5]
4.2.2. Find E(X) and Var(X) using the characteristic function of X.
[8]
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QUESTION 5 [17 marks]
5.1. Let X, and X2 be independent random variables with the joint probability density function given
by
fA,X4,X2) %2)
_= fe
—(x1+X2)
0,
,
iiffotxh1er>wi0;s.e4 .x2 > 0,
Find the joint probability density function of Y, = X, + Xz, andY, = ee
[10]
1
2
5.2. Let X and Y be independent Poisson random variables with parameters A, and Az. Use the
convolution formula to show that X + Y isa Poisson random variable with parameter A, + Az.
[7]
=== END OF PAPER===
TOTAL MARKS: 100
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