MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 02 (Suggested Answers)
1. (a)
D
60
70
130
D
30
20
50
T
T
Total
Total
90
90
180
(b) P (T) = 90/180 = 0.5
(c) P (D) = 50/180 = 0.2778
(d) P (D
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 04 (Suggested Answers)
1. Let
X:
height of a randomly selected child
XN : height of a normal child
XA : height of a DA child
XB : height of a
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 01 (Suggested Answer)
1. (a) mean: 63.3750, standard deviation: 22.6270, variance: 511.9821
(b) mean: 45.1111, standard deviation: 31.2428, v
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 03 (Suggested Answers)
1. (a) f (3) = 0, f (4) = 1/5, f (5) = 4/5, and f (x) = 0 otherwise.
f (x) 0 for all x.
P
f (x) = 0 + 1/5 + 4/5 = 1.
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 02
1. The followings are some information about a study of the effectiveness of an indicator of
Disease D.
there are 180 subjects in the stu
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 04
1. In country X, two diseases (DA and DB) affect the statures of children seriously
when they are 10-year-old. It is known that the height
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 01
1. Find the mean, standard deviation, and variance of the following datasets:
(a) 36, 84, 59, 55, 92, 29, 76, 76
(b) 39, 57, 8, 6, 54, 78,
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Exercise 03
1. Consider the random variable X whose probability distribution is given by
f (x) =
(x 3)2
5
for x = 3, 4, 5.
(a) Verify that this is a v
Chapter 5
Integer Programming
1
1.1 Introduction to Integer Programming
Simply stated, an Integer Programming (IP for
short) is an LP in which some or all of the
variables are required to be non-negative
integers.
Many real-life situations may be formul
MATH2005 Probability and Statistics for Computer Science
MATH2006 Probability and Statistics for Science
Assignment 3
Due date: 30 March 2017
(Reminder: Keep a soft copy of your answers of the assignment
before you submit the hard copy to assignment box)
Chapter 5
Integer Programming
1
1.1 Introduction to Integer Programming
Simply stated, an Integer Programming (IP for
short) is an LP in which some or all of the
variables are required to be non-negative
integers.
Many real-life situations may be formul
Chapter 4
Duality Theory and Sensitivity Analysis
Also see Chapter 6 of the text book.
One of the most important discoveries in the early development of linear programming
was the concept of duality and its many important ramifications. This discovery rev
Solutions to the EndofChapter Excereises in
InteriorPoint Algorithm: Theory and
Analysis
Yinyu Ye
November 1997 1997
Yinyu Ye7 Professor
Department of Management Sciences
The University of Iowa
Iowa City, IA 522421000
Phone: (319) 3351947
Email: [email protected]
2 Seeking Feasibility in Linear Programs
A general linear program has the form cfw_min, max cx, subject to Ax cfw_, , = b,
l x u, where c is a 1 n row vector, x, l, u, and b are n 1 column vectors, and
A is an m n array, all consisting of real numbers. It
1
Duality Theory
Recall from Section 1 that the dual to an LP in standard form
cT x
Ax b, 0 x
maximize
subject to
(P)
is the LP
bT y
AT y
minimize
subject to
(D)
c, 0 y.
Since the problem D is a linear program, it too has a dual. The duality terminology
s
DM545
Linear and Integer Programming
Lecture 7
Revised Simplex Method
Marco Chiarandini
Department of Mathematics & Computer Science
University of Southern Denmark
Outline
Revised Simplex Method
Efficiency Issues
1. Revised Simplex Method
2. Efficiency Is
0.1
Example of Simplex Method (Two Phases):
Consider the following example: min z : x 0 and
2x1 + x2 + 2x3 + x4 + 4x5
4x1 + 2x2 + 13x3 + 3x4 + x5
x1 + x2 + 5x3 + x4 + x5
= z
= 17
= 7
After introducing artificial variables and getting initial canonical for
Chapter 1
Introduction to Linear Programming
Also see Chapter 3 of the text book.
1
History and Application of Optimization
In mathematics, computer science and operations research, mathematical optimization (alternatively, mathematical programming or sim
Name:_alternate_solution
ESE 403
Operations Research
Fall 2010
Examination 2
Closed book/notes/homework/cellphone examination. You may use a calculator. Please
write on one side of the paper only. Extra pages will be supplied upon request.
You will only r
Chapter 2
The Simplex Method
Also see Chapter 4 of the text book.
1
Also see Chapter 4 of the text book.
1
Basic Concept
Terminology
1 1.1Basic
Concept
illustrate
basic
concept
considering
following
problem:
WeWe
illustrate
thethe
basic
concept
byby
consi
HONG KONG BAPTIST UNIVERSITY
Page: 1
SEMESTER 1 MID-TERM EXAMINATION, 2016-2017
Course Code: MATH3205
Section No.: 00001
Course Title: Linear and Integer Programming
Time Allowed:
of 17
Hour(s)
Total Number of Pages:
17
INSTRUCTIONS:
1. Answer ALL of the
Linear Programming and Integer Programming
Assignment 2
November 8, 2016
Submit on or before November 16 2016 to my letter box at Floor 12, FSC Building
1. Consider the following problem.
max
Z = x1 2x2 x3
x1 + x2 +2x3 12
s.t.
x1 + x2 x3 1
x2 0,
x3 0.
x1
MAXIMIZATION & MINIMIZATION WITH MIXED PROBLEM CONSTRAINTS
The Big M Method
Form the modified problem
Step 1: If any problem constraints have negative constants on the right side, Multiply both sides by 1
to obtain a constraint with a nonnegative constant
MATH 407 Key Theorems
Theorem 0.1 (Weak Duality Theorem). If x 2 Rn is feasible for P and y 2 Rm is feasible for D, then
cT x y T Ax bT y.
Thus, if P is unbounded, then D is necessarily infeasible, and if D is unbounded, then P is necessarily
infeasible.