Black holes
Black holes
Black holes in Newtonian theory; critical radius
Black holes in General Relativity; gravitational redshift
Schwarzschild black hole
Formation of black holes
gravitational time dilation, tidal force, Hawking
Radiation
Searching f
Cosmology
()
Cosmology
The Cosmological Principle
Edwin Hubbles discoveries
General relativistic cosmology
(Cosmic microwave background and the Big Bang)
(dark energy)
(dark matter)
Some examples of current research in Cosmology
Dark matter
Scie
Cosmology
()
Cosmology
The Cosmological Principle
Edwin Hubbles discoveries
General relativistic cosmology
(Cosmic microwave background and the Big Bang)
(dark energy)
(dark matter)
Fate of the universe
: (A)(B)
(C) Three possibilities: Open
(A)
Cosmology
()
Cosmology
The Cosmological Principle
Edwin Hubbles discoveries
General relativistic cosmology
(Cosmic microwave background and the Big Bang)
(dark energy)
(dark matter)
(The Cosmological Principle )
~
Any observer anywhere in the u
UGEB2341 Relativity and Quantum Mechanics
Due Novebmer 29, 2011
Assignment 6
1. A spaceship is traveling at 0.4 c from Earth to a star 2 light years away, and two laser
pulses are fired from it towards Earth and the star when it is half way there. Which o
UGEB2341 Relativity and Quantum Mechanics
Due November 15, 2011
Assignment 5
1. Read my article Ch. 6 (downloadable from course webpage) and Ch. 5 of The
Elegant Universe. The author explained the incompatibility between general
relativity and quantum mec
UGEB2341 Relativity and Quantum Mechanics
Assignment 4
due November 1, 2011
1. Read my article Ch. 5 (downloadable from course webpage), and Greene The
Elegant Universe p. 85116 (Ch. 4). The author discussed Schrdingers suggestion
that the waves causing
UGEB2341 Relativity and Quantum Mechanics
Due October 18, 2011
Assignment 3
1. Read my articles Ch. 3 and Ch. 4 (downloadable from course webpage), and
Greene The Elegant Universe p. 7884 (second half of Ch. 3) and p.345356
(first half of Ch. 14). The a
UGEB2341 Relativity and Quantum Mechanics
Due October 11, 2011
Assignment 2
1. Read my article Ch. 2 (2.5 2.8) (downloadable from course webpage), and
Greene The Elegant Universe p. 4678 (second half of Ch. 2 and first half of Ch. 3).
2. In Ch. 2 of The
UGEB2341 Relativity and Quantum Mechanics
Assignment 1
due 27/09/2011
Read my articles Ch. 1 and Ch. 2 (2.1 2.4) (downloadable from course webpage),
and Greene The Elegant Universe p. 2346 (first half of Ch. 2).
1. Point out two conceptual differences be
STA 2003 Tutorial 12
27/11/2005
Monte Carlo Methods (Simulation)
Example 1
We want to approximate the following integral:
1
0
1 x 2 dx
Notice: This integral is simply E( 1 x 2 ) where x ~ Uniform(0,1)
Algorithm:
1. Generate 1000 random points from Uniform
STA 2003 Tutorial 8
27/11/2005
Monte Carlo Methods (Simulation)
1. They are methods that provide approximate solutions to mathematical
problems by performing statistical sampling experiments on a computer.
2. It is a very powerful alternative to the tradi
STA 2003 Tutorial 7
1/11/2005
What is Computer Simulation?
A computer simulation is the process of designing a mathematical or logical model of
a real system and then conducting computerbased experiments with the model to
describe, explain, and predict t
STA 2003 Tutorial 10
14/11/2006
Statistical Graphics
(B) Pie Charts
 They are used for qualitative variables.
 The area of each sector relative to the total area represents the total units in a
category.
 Area of the whole circle represents the total.
STA 2003 Tutorial 10
14/11/2006
Example 1:
 The thickness and tilting of the pie give a wrong perception that the sectors
in the front are larger.
 There are too many sectors making the graph too complicated.
 The total and percentage of each sector ar
Counts of unique variable
Color
Blue
Red
Red
Blue
Red
Red
Blue
Density
Hi
Lo
Lo
Hi
Hi
Lo
Lo
Simple
Cluster
Stack
Counts of Color
Counts of color by
Density
Counts of color by
Density
Function of a variable
One Y
Color
Blue
Red
Blue
Red
Blue
Blue
Density
H
STA 2003 Tutorial 8
31/10/2006
Features of a good graph
(a) Perceptual accuracy: Make readers get a correct perception of the
information. Experiments show that in terms of perception, graphical elements
are ordered from most accurate to least accurate as
Review of past exercises
1.Classic School problem with graphic aided.
(1)Two points L and M are placed independently and uniformly on a segment AB.
Find the probability that the point L is closer to M than to A.
(2) Example 1.7: A bus of line A arrives at
STA 2003 Tutorial 6
17/10/2005
Measure of Spread
A numeric value is a valid spread measure if
(a) It has a meaningful zero. It is zero if, and only if all data values are exactly the
same.
(b) It takes nonnegative value. The smaller the value is, the more
Measures of Central Tendency

It is known as measure of center or location, it can be defined as a value of m
which minimizes
n
d ( x , m) for a distance function d(x,y)
i =1
i

The location measure is required to have the following properties:
Measure
STA 2003 Tutorial 4
03/10/2006
In data analysis, we usually classify values as either qualitative or quantitative.
Qualitative value is value which acts as a label of category.
Quantitative value is a real number, and arithmetic operations are permitted.
hk3rd tutorial
1. Maximum Likelihood Estimate (MLE)
Given the outcome data, we can estimate the value of parameter.
And our criterion here is that:
We choose the value of parameter that makes the probability of
outcome the greatest one compared to all oth
STA 2003 Tutorial 2
19/9/2006
Basic concepts:
(1) Basic procedures of statistical analysis
 Designing the data collection procedure
 Collecting data
 Summarizing data
 Presenting data
 Analyzing data
 Results Interpretation and Decision Making
(Plea
STA 2003 Tutorial 1
12/9/2006
I. Calculating Probabilities for Equally Likely events
Pr (an event A) = (number of favorable outcomes) / (number of all possible outcomes)
e.g. A die is rolled once.
 All possible outcomes are: cfw_1, 2, 3, 4, 5, 6
 Number
STA 2003 Computeraided Statistical Reasoning
1st term, 2005/2006
Test 2
Name:
Student ID number:
Part I: Multiple choices (40%):
1. Which of the following graphs is the weakest in perceptual accuracy?
(a) Bar chart
(b) Pie chart
(c) Line graph
(d) Scatte
Some amendment of 3rd tutorial notes
1. Hypothesis testing in lecture notes
A testing procedure is good if P(type I error) and P(type II error) are small. Usually
we can decrease one error probability only at the expense of increasing another error
probab
STA 2003 Computeraided Statistical Reasoning
1st term, 2006/2007
Problems (Chapter 2)
1.
(a) Demonstrate the difference between ordinal scale and interval scale using
examples.
(b) Classify the following data into one of the following four measurement le
1st term 2006/2007
STA 2003 Computeraided Statistical Reasoning
Final Examination:
(a) It is a Closed Notes Examination
(b) No calculator is allowed.
(c) When you are asked to give a defect of a statistical graph, give the most serious
defect. You cannot
Computing Session 7: Monte Carlo estimation of e
Let U1, U2, be a sequence of independent U(0,1)
random numbers. Define X to be the smallest integer
such that UX1 > UX. Prove that E(X) = e.
Proof: (We can ignore the possibility that any two
U(0,1) random