Mathematical Probability 1 (MATS2105)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2014
REPUBLIC OF CAMEROON REPUBLIQUE DU CAMEROUN
Peace-Work-Fatherland Paix-Travail-Patrie
MINISTER OF HIGHER EDUCATION MINISTERE DE L’ENSEIGNMENT
THE UNIVERSITY OF BAMENDA
UNIVERSITE DE BAMENDA
FACULTY OF SCIENCE
FACULTE DES SCIENCE
P.O. Box 39 Bambili
Tell: (+237) 22 81 63 50
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE.
FIRST SEMESTER EXAMINATION
SEMESTER: 1 ACADEMIC YEAR: 2013-2014
COURSE CODE: MATS2105 COURSE TITLE: MATH. PROBABILITY 1
TIME ALLOWED: 3 Hours.
Answer all questions. ’
1.
What is -algebra of subsets of a given set Ω ?-2 pts
2.
What is a probability space? 3pts
3.
Let Ω be a set and be a -algebra of subsets of Ω. What is a random variable relative to ? 3 pts
4.
When are events A, B,C independent relative to a probability measure P? 3 pts
For questions 6 to 9, write down only the correct answer.
5.
A communication system is to consist of n seemingly identical antennas that are to be lined up in a
linear order. Suppose that m of the antennas are defective.
(i) In how many ways can the antennas be lined up so that no two adjacent antennas are defective
and the first and last antennas are not defective? 2 pts
(iij What is the maximum of the values of m for which such an arrangement is possible? 2pts
6.
A community consists of 10 men each of whom has 3 sons. If one man and two of his sons are to be
chosen as father a and son of the year,
(i)
how many different choices are possible? 2pts
(ii)
in how many ways can these choices be made ? 2pts
7.
From a group of 5 men and 7 women, how many different committees consisting of 2 men and 3
women can be formed ? 2pts
8.
A fair coin is tossed four times. Find the probability of tossing
(a)
at least three heads 2pts
(b)
exactly three heads 2pts
(c)
three or more heads consecutively 2pts
(d)
three heads consecutively 2pts
9.
In order to start playing a game of chance with a die, it is necessary that you first toss a six.
(a) What is the probability that you toss your first six at your third attempt ? 2 pts
(b)
What is the probability that you require more than three tosses to get your first six? 2 pts
(c)
After how many tosses, would your probability of having tossed a 6 be 0.95? 2 pts(the tosses are
independent)
For questions 10 to the end, show all necessary working.
10. A coin which lands tails with probability
is tossed 3 times. Assume that the tosses are independent and let
£ be the number of heads observed. Write down the distribution and distribution function of £. 2+5 pts
11. A random variable has an exponential distribution with parameter . Write down the density and the
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distribution function of . 2+3 pts
12. The distribution function of a random variable £ is given by F(x)=
(i) Write down the distribution of £. 3 pts
(ii) find £
. 6 pts
13. A random variable f, takes values in the set {—2, —1,7 0, 1, 2} such that P(£=1)=P(£=-1) and P(£=2)=P(£=-
2)=q, where 0<p<1 and 0<q<1. Write down the distribution function max (-1, £). 5 pts.
14. the joint density of the random vector (£,) is given by f(x,y)=
!∞
!∞
"#$%&'(
. find each of the
following:
(i) P(£>1, )). 2 pts
(ii) P(£ ) 4pts
(iii) P()***3pts
15. Let X and Y be independent random variables and suppose that X has a Poisson distribution with parameter
> 0 and Y has a Poisson distribution with parameter +> 0. Prove that X + Y has a Poisson distribution with
parameter + +. 5 pts
Happy Christmas in Advance !
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