13
Bernoulli Experiment and Its Related Distributions
An experiment is called a Bernoulli experiment (or Bernoulli trial) if there
are only two possible outcomes: success with probability p and failure with
probability (1 p), where 0 < p < 1.
We say X is

MATH 1853
Problem Set #3
1.
1 4 1
Find the eigenvalues and eigenvectors of the matrix A 2 1 1 .
3 1 2
2.
7 4 1
It is known that A 4 7 1 has eigenvalues
4 4
3,3,12 , find the value
of and the eigenvectors of A .
3.
Suppose the 3-by-3 matrix A has 3 ei

MATH 1853
Problem Set #4
1.
1 2 0
Given A 2 2 0 , x
0 0 1
3
and x 1 .
(a) Decompose A into A QQT where Q is an orthonormal matrix and
is a diagonal matrix containing the eigenvalues of A . Then, decide if
A is positive definite, negative definite or

MATH 1853
Problem Set #1
1.
Solve the following two equations using Gaussian elimination:
2 x1 3x2 11x3 5 x4 2
x x 5x 2 x 1
1 2
3
4
2
x
x
3
x
2
x
3
4 3
1 2
x1 x2 3x3 4 x4 3
and
4 x1 x2 x4 4
x 4x x 1
1
2
3
x
4
x
x
3
4 4
2
x1 x3 4 x4 10
2.
1 2 3 4

MATH1853 Maths I
Eigenvalues & Eigenvectors
Dr. Ngai WONG
Fall 2016
1
Eigenvalues and Eigenvectors
For an nn matrix A, if there is a nonzero vector x such that Ax=lx for
some scalar l, then l is an eigenvalue of A, and x is the eigenvector
corresponding t

MATH1853 Maths I
Inner Product
Dr. Ngai WONG
Fall 2016
1
Inner Product
For two vectors u, v in n, the inner product is defined as
v1
v
uT v u1 u2 . un 2 u1v1 u2v2 . un vn
vn
It is obvious that uTv= vTu
From inner product, we can define other at

MATH1853 Linear Algebra, Probability and Statistics Homework 2
Due Date: 28 Oct 2016 (Fri) 5pm. Hardcopy Only. Please put into HW collection boxes
(labeled with MATH1853) at Chow Yei Ching Bldg 7th floor near the Lab CYC711.
1.
4 4 2
What is the rank of

MATH1853 Maths I
System of Linear Equations
Dr. Ngai WONG
Fall 2016
You MUST attend lectures
Reasons:
1. (disclosed during lecture)
2. (disclosed during lecture)
3. (disclosed during lecture)
4. (disclosed during lecture)
5. Notes + self-study + intellig

MATH1853 Maths I
Determinant
Dr. Ngai WONG
Fall 2016
1
Determinants
1 d b
Recall the 22 matrix inverse equation A
ad bc c a
1
Is it possible to extend the result to matrix of dimension nn ?
We first look at the concept of determinant
a11 a12
A
For a

MATH 1853
Problem Set #2
1.
4
5 2
1 0 2
Given A 6 4 0 and B 3 1 2 .
1 4 3
6 2 1
Find A-1 and B-1. Verify that (AB)-1 = B-1A-1.
2.
1 2 3
Find all the cofactors of the determinant 1 0 1 . Hence evaluate the determinant.
1 1 1
3.
ab ac
Compute the determina

Definition of the Cross Product:
u v ( u v sin )n.
( now, it is a vector! )
The new vector n (the direction of the cross product) is perpendicular
to u and v. It is closely associated with the right-hand screw rule.
The construction of u v can be illustra

Example 1. Consider the function
cfw_ 2
3x if 0 x 1,
p(x) =
0 otherwise.
Determine if it is a probability density function. If yes find P (X 0.5).
1
We note that p(x) 0 for any x [0, 1]. Secondly, we see that
1
p(x)dx =
3x2dx = 1.
0
Therefore p(x) is a

MATH1853 II T3
Complex variables:
roots of unity, complex functions
F. Tsang
University of Hong Kong
March 29, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T3
March 29, 2016
1 / 11
Root of unity
How do we solve a polynomial equation
x 3 + ax 2 + bx

Vectors
A real number is a point on the real line
To describe a point on a plane 2, we use two numbers, e.g., (3,-1)
In fact, this is just an expression of a point using rectangular
coordinate system
3
If we put the coordinates as , we have a vector
1
0

Determinants
1 d b
ad bc c a
Recall the 22 matrix inverse equation A 1
Is it possible to extend the result to matrix of dimension nn ?
We first look at the concept of determinant
a11 a12
For 22 matrix A a a , its determinant is det(A)=a11a22-a12a21
2

Eigenvalues and Eigenvectors
For an nn matirx A, if there is a nonzero vector x such that Ax=lx
for some scalar l, then l is an eigenvalue of A, and x is the
eigenvector corresponding to l
If we view A as a mapping, Ax=lx means that the mapping A acting
o

MATH1853 II T5
Random variables
F. Tsang
University of Hong Kong
April 15, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T5
April 15, 2016
1 / 12
Random variables
Recall: A sample space S is the collection of all possible outcomes of an
experiment.

MATH1853 II T6
Geo(p), Bin(n, p), Poi(), exp()
F. Tsang
University of Hong Kong
April 21, 2016
F. Tsang (University of Hong Kong)
MATH1853 II T6
April 21, 2016
1 / 12
Geometric random variables (Beginner)
Experiment: shooting at a target until there is a

MATH 1853 Homework (Spring 2016)
1. Determine the values of k so that the following system in unknowns x, y, z has: (i) a
unique solution, (ii) no solution, (iii) an infinite number of solutions:
x 2y 1
x y kz 2
ky 4 z 6
2. Let W be the solution space of

MATH1853 Maths I
Matrix/Matrices
Dr. Ngai WONG
Fall 2016
1
Matrices
As seen in previous notes, a matrix consists of vectors as columns:
A=[a1 a2 an]
For an mn matrix, its structure is
Two matrices are equal iff they have the same size and their
corresp

MATH1853 Maths I
Vectors
Dr. Ngai WONG
Fall 2016
1
Vectors
A real number is a point on the real line
To describe a point on a plane 2, we use two numbers, e.g., (3,-1)
In fact, this is just an expression of a point using rectangular or
Cartesian coord

MATH1853 Maths I
Eigenvalues & Eigenvectors
n. [ ]
[ ]
[ ]
Dr. Ngai WONG
Fall 2016
1
Eigenvalues and Eigenvectors
For an nn matrix A, if there is a nonzero vector x such that Ax=lx for
some scalar l, then l is an eigenvalue of A, and x is the eigenvector

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AS/1853/MATH1853/4
UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1853
Assignment 4 Normal Distribution and Statistical Inference
Due Date: April 30, 2016 (Saturday) (5:00pm)
Please submit your assignment through Moodle. No late submission will be

MATH 1853 Homework (Spring 2017)
1. Determine the values of k so that the following system in unknowns x, y , z has: (i) a
unique solution, (ii) no solution, (iii) an infinite number of solutions:
x 2y 1
x y kz 2
ky 4 z 6
2. Let W be the solution space of