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PBT602S - PROBABILITY THEORY 2 - 2ND OPP - JULY 2022 |
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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: JULY 2022
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY / SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
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.
4. 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. Briefly explain the following:
1.1.1. Boolean algebra B(S)
[2]
1.1.2. Measure on a B(S) algebra
[2]
1.2. Show that if m is a measure on B(S) and c = 0, then cm is a measure, where (cm)(A) =
c.m(A)
[4]
1.3. LetS = {1,2,3}, then find:
1.3.1. Power set, P(S)
[2]
1.3.2. size of P(S)
[2]
Question 2 [17 marks]
2.1. An insurance company offers its policyholders a number of different premium payment options.
For a randomly selected policyholder, let X be the number of months between successive
payments. The cumulative distribution function of X is
0,
0.4,
FG) = {05
0.8,
la,
ifx <1,
ifl<x<3,
if 3<x<5,
ifS5<x</7,
ifx > 7.
2.1.1. Use F(x) to compute P(3 < X <5).
[2]
2.1.2. Find is the probability distribution function/probability mass function of X?
[3]
2:2: Let X and Y be a jointly distributed continuous random variable with joint p.d.f. of
6
fires) = fe 4”
0,
forO<x<1;0<y1,
otherwise.
2.2.1. Show that marginal p.d.f. of X, fy(x) = s(x +5) Io, 1) (%)
[2]
2.2.2. Find P(Y = 0.15|X = 0.25)
[5]
2.3. Suppose that the joint p.d.f. of two continuous random variables X and Y is given by
_ (12x, 0O<y<x<1; 0<x?<y<1,
fer y) = 0, elsewhere.
Find the marginal p.d.f. of Y.
[3]
2.4. The average weight of individuals in city A was 95kg with standard deviation of 10. If the city
contains 100, 000 residents, what is the minimum number of individuals with a weight
between 70kg and 120kg?
[2]
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QUESTION 3 [30 marks]
3.1.If X and Y are linearly related, in the sense that Y = aX + b, where a > 0, then show that
Pxy = 1.
[5]
3.2. Let X,, Xz, .... Xp, be independently and identically distributed with normal distribution with
mean p and variance o”. Then show, using the moment generating function, that Y =
xX
has a normal distribution and find the mean and variance of Y?
Hint: My,(t) = ett t2o-2
[8]
3.3. Find the cumulant generating function for X~N(, 07) and hence find the first cumulant and
the second cumulant.
[7]
3.4. Let the random variable X~N(u, 07). Find E(X) and Var(X) using the characteristic function
of X. HINT: $y (t) = el° t#-13°*
[10]
QUESTION 4 [26 marks]
4.1. Acertain radioactive mass emits alpha particles from time to time. The time between emissions,
in seconds, is random, with probability density function
_ (0.5e-95”, fory> 0,
fr)
{0
otherwise.
4.1.1. Find the 25" percentile of the time between emissions
[3]
4.1.2. Find the median time between emissions
[3]
4.2. If X is a random variable having a binomial distribution with the parameters n and p (i.e.,
X~Bin(n, p)), then
3.4.1. Show that the moment generating function ofX is given by My(t) =
(1 — p(1 —e*))”. Hint: (a + b)” = Yieo(,)aXb™*
[4]
3.4.2. Find the first moment about the origin using the moment generating function of X. [3]
4.3. Let random variables X,~Poisson(A,) for k = 1,..., m1 be independent Poisson random
variables. If we define another random variable Y = X; + Xz +---+Xpy, then find ¢y(t).
Comment on the distribution of Y based on your result. [Hint py, (t) = erk(e"—-1)]
[7]
4.4. Let Y be continuous random variable with a probability density function f(y) > 0. Also, let
U = h(Y). If his increasing on the range of a given random variable, then show that
[6]
fulw) = fr(n-2(Snu))(u)
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QUESTION 5 [15 marks]
DL. Let X, and X, have joint p.d.f. f(x,, x2) = 2e~%1**2) for 0 <x, <x <1. Let ¥, =X, and
Y, = X, + X2. Find the joint p.d.f. of ¥, and Yj, g(v1, y2)
(9]
5.2. Suppose that X and Y are independent, continuous random variables with densities fy (x) and
fy(y). Then Z = X + Y is a continuous random variable with cumulative distribution function
frsy(2) = [ fe-— lfyz(dy
(6]
=== END OF PAPER===
TOTAL MARKS: 100
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