MA2101
Linear Algebra II
AY 2004/2005 Sem 2
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Lau Tze Siong
MA2101 Linear Algebra II
AY 2004/2005 Sem 2
SECTION A
Question 1
(i) For all
a1 b1
a1 a1
a2 b2
a2 a2
,
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER 2 EXAMINATION 2011-2012
MA2213
Numerical Analysis I
April 2012 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper consists of TWO (2) sections: Section A and Sect
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER 1 EXAMINATION 2011-2012
MA2213
Numerical Analysis I
November 2011 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper consists of TWO (2) sections: Section A and S
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER EXAMINATION FOR THE DEGREE OF B.SC.
SEMESTER 1 EXAMINATION 20122013
MA2213
Numerical Analysis I
December 2012 Time allowed : 2 hours
Instructions to Candidates
1. This examination paper c
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER EXAMINATION FOR THE DEGREE OF B.SC.
SEMESTER 1 EXAMINATION 20082009
MA2213
NUMERICAL ANALYSIS I
November 2008 Time allowed : 2 hours
Instructions to Candidates
1. This examination paper c
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF MATHEMATICS
SEMESTER EXAMINATION FOR THE DEGREE OF B.SC.
SEMESTER 2 EXAMINATION 20102011
MA2213
Numerical Analysis I
April 2011 Time allowed : 2 hours
Instructions to Candidates
1. This examination paper cont
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 7
Suggested Solutions
1. Let N denote the number of games played.
1
(a) E (N ) = 2[p2 + (1 p)2 ] + 3[2p(1 p)] = 2 + 2p(1 p) which is maximized when p = .
2
3
3
2
2
2
2
(b) E [N ] = 3[p + (1 p) ] + 4
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 6
Suggested Solutions
(i) Prob = P (1 up,1 down) + P (1 down, 1 up) = 2p(1 p).
1.
(ii) Prob =P ( 2 ups, 1 down) =
32
p (1 p) = 3p2 (1 p).
2
(iii)
P ( up on 1st day | in 3 days, up 1 unit )
P ( up on
N ATIONAL U NIVERSITY O F S INGAPORE
EXAMINATION
ST2131/MA2216
(SEMESTER
PROBABILITY
2: A Y 2 009-2010)
April/May 2010 -
Time allowed: 2 h ours
INSTRUCTIONS T O CANDIDATES
L This examination paper contains SIX (6) questions and comprises NINETEEN (19
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 2 EXAMINATION 2008-2009
MA2213
Numerical Analysis I
May 2009 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination.
2. This examination paper contains a total of
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2010-2011
MA2213
Numerical Analysis I
November 2010 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination. Each student is allowed to bring one pie
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2011-2012
MA2216/ST2131
Probability
November 2011 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper consists of FOUR (4) questions and comprises
FOUR (4)
N ATIONAL U NIVERSITY O F S INGAPORE
EXAMINATION
ST2131/MA2216
(SEMESTER
PROBABILITY
2: A Y 2 009-2010)
April/May 2010 -
Time allowed: 2 h ours
INSTRUCTIONS T O CANDIDATES
L This examination paper contains SIX (6) questions and comprises NINETEEN (19
MA2216/ST2131
Probability
AY 2009/2010 Sem 2
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Agus Leonardi
MA2216/ST2131 Probability
AY 2009/2010 Sem 2
Question 1
(a) Let X (, ).
MX (t) = E etX
etx
=
0
ex (x)
MA2216/ST2131
Probability
AY 2010/2011 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
Luo Xuan, Ebenezer Tan
with credits to Tan Ah Kow, Koh AhLee
MA2216/ST2131 Probability
AY 2010/2011 Sem 1
Question 1
(a) Since f (x
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2012-2013
MA2216/ST2131
Probability
November 2012 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This examination paper consists of FOUR (4) questions and comprises
THREE (3)
MA2216
Probability
AY 2011/2012 Sem 1
NATIONAL UNIVERSITY OF SINGAPORE
MATHEMATICS SOCIETY
PAST YEAR PAPER SOLUTIONS
with credits to Alan Chee
MA2216 Probability
AY 2011/2012 Sem 1
Question 1
yx
x
(i) If u = xy , v = x/y , then J (u, v ) = 1 x = 2 = 2v
y
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 1 EXAMINATION 2009-2010
MA2213
Numerical Analysis I
November 2009 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination. Each student is allowed to bring one pie
NATIONAL UNIVERSITY OF SINGAPORE
FACULTY OF SCIENCE
SEMESTER 2 EXAMINATION 2009-2010
MA2213
Numerical Analysis I
April 2010 Time allowed : 2 hours
INSTRUCTIONS TO CANDIDATES
1. This is a closed book examination. Each student is allowed to bring one piece
NATIONAL UNIVERSITY OF SINGAPORE
DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY
ST2131/MA2216 PROBABILITY
SEMESTER II, AY2012/2013
Tutorial 10
1.
2.
The joint density of the point ( , ) at which the accident occurs is
1
3
33
3
, (, ) = ,
for < < , < <
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 8
Suggested Solutions
1. (a) The probability that an arbitrary couple were both born on April 30 is, assuming
independence and an equal chance of having being born on any given date, (1/365)2 .
Henc
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 6
Exercise 1. A simplied model for the movement of the price of a stock supposes that on each
day the stocks price either moves up 1 unit with probability p or moves down 1 unit with
probability 1 p
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 5
Exercise 1. Consider 3 urns. Urn A contains 2 white and 4 red balls; urn B contains 8 white
and 4 red balls and urn C contains 1 white and 3 red balls.
(a) Select one ball from a randomly chosen u
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 7
Exercise 1. Suppose that two teams play a series of games that ends when one of them has won
i games. Suppose that each game played is, independently, won by player A with probability
p. Find the
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 8
Exercise 1. Approximately 80, 000 marriages took place in the state of New York last year.
Estimate the probability that for at least one of these couples
(a) both partners were born on April 30;
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 9
Exercise 1. A point is chosen at random on a line segment of length L. Interpret this statement
1
and nd the probability that the ratio of the shorter to the longer segment is less than .
4
Exerci
Tutorials for ST2131/MA2216, Probability, 2012/2013
Week 4
Exercise 1. A fair die is rolled twice in sequence. Assume that all outcomes are equally
probable. What is the probability that the second roll lands on a strictly higher value than the
rst does?