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PBT602S- PROBABILITY THEORY 2 - JAN 2020 |
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e
NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
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: JANUARY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SECOND OPPORTUNITY/SUPPLEMENTARY EXAMINATION QUESTION PAPER
EXAMINER
Dr. D. NTIRAMPEBA
MODERATOR: | Dr. D. B. GEMECHU
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations.
3. All written work must be done in blue or black ink and sketches must
be done in pencil. Marks will not be awarded for answers obtained
without showing the necessary steps leading to them
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
ATTACHMENTS
1. None
THIS QUESTION PAPER CONSISTS OF 2 PAGES (Excluding this front page)
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Question 1 [20 marks]
1.1 Briefly explain the following terminologies as they are applied to probability theory:
(a) Boolean algebra B(S)
[2]
(b) o algebra
[3]
c ) Measure on a B(S) algebra
[3]
d) Convolution of two integrable real-valued functions f and g
[3]
1.2 Let S = {a,b,c,d}. Find:
(a) P(S),
(b) size of P(S).
[(14]]
1.3 Consider the random variables X and Y that represent the number of vehicles that arrive at
two separate corners during a certain 2-minute period. These two street corners are fairly
close together so that. it is important that the traffic engineers deal with them jointly if
necessary. The joint distribution of X and Y is known to be
f(z,y) = { p9a1
, z= 0,1,2,...,9=0,1,2,...,
, otherwise
Find the probability that less than 4 vehicles arrive at the two street corners during the
stated time period.
[4]
Question 2 [30 marks]
2.1 Let X and Y denote the lengths of life, in years, of two components in an electronic system.
If the joint density function of these variables is
_ e —(r+y)
f(z,y) = ‘ 0
,ct> 0, y > 0,
, otherwise,
then find the mean value of Y.
[6]
2.2 Suppose X and Y are random variables such that (X,Y) must belong to the rectangle in
xy- plane containing all points (x, y) for which 0 < x < 3 and 0 < y < 4. Suppose that the
joint cumulative distribution of X and Y at any point (z, y) in this rectangle is specified as
follows: F(z,y) = seule tu)
(a) Use the joint cumulative distribution, F(z, y), to find (P(1 <2 <2,1<y< 2)
[7]
(b) Find the joint probability density function of X and Y
[4]
2.3 Let X be a discrete random variable with mean p and variance g?. Also, let k be some
positive integer. Show that P[|X — p| < ko] >1-.
[13]
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Question 3 [20 marks]
3.1 Let X be a random with a probability density function f(x) and a moment-generating func-
tion denoted by m x(t). Show that mx(t) packages all moments about the origin in a single
expression. That is, mx(t) = ~32c.o9 Gtki.
[5]
3.2 Let X be a random variable whose moment-generating function, denoted by m x(t), exists.
Show that its second cumulant (k2) is related to its first and second moments by the following
relationship ky = [2 — p?.
[5]
3.3 (a) Show that the cumulant-generating function of an exponential random variable (X), with
a mean +, is Kx(t) = Ind — In[-r¢].
[5]
(b) Use the cumulant-generating function provided above to find the variance of X.
[5]
Question 4 [30 marks]
4.1 The waiting time, in hours, between successive speeders spotted by a radar unit is a contin-
uous random variable with cumulative distribution function
1—e** forx>0
Me} = { 0
otherwise ,
Derive the characteristic function of X and use it to find the mean of X.
[8]
4.2 Let Y be continuous random variable with a probability density function f(y) > 0. Also, let
U =h(Y). Then show that
fulu) = fre (adyh~*,(u)
iu
4.3 Suppose that X, and X2 have a joint pdf given by f (x1, 22) = 2e~+) for 0 < a < 2_ < 0.
Let Y; = X; and Yo = X, + Xo.
(a) Find f(y, ya);
(10]
(b) Find f(y);
[2]
(c) Find f (ye);
[2]
(d) Are Y; and Y> independent?
(1]
END OF QUESTION PAPER