Embed Size (px)
DESCRIPTION
lecture 1 of probablity
Citation preview
z. zheng 1
Probability and Statistics
Prof. Zheng Zheng
教育部来华留学英语授课品牌课程
z. zheng 2
Course Outline
• Named as a National Quality Curriculum Taught in English for International Students by the Ministry of China
• 18 class lectures: Sept. 25, 2014-Jan. 22, 2015
• Time: Thursday 9:00-11:45 am
• Many sessions will be video-recorded this semester
– 1 midterm (~1st week of Dec., TBD) and 1 final (last week of Jan. 2015, TBD)
z. zheng 3
Lecturers
• Experienced Teaching Team (3 Full: – Prof. Zheng Zheng [email protected] (Probability)
– Prof. Huaping Xu [email protected] (Stochastic)
– Prof.Yichuan Yang [email protected] (Statistics)
– Lecturer.Xin Zhao [email protected] (Probability)
• Teaching Assistants/ Office Hour:
- Liya Shu: email: [email protected] 15652915285
- Guangnan Chen: email: [email protected] 13126732863
- Address: New Main building F703, Monday 19:00-20:00
z. zheng
Important Course Information (1/3)
• Course website:
http://www.ee.buaa.edu.cn/oldeeweb/html/eng
lish/common/Excellent%20course.html
• Lecture notes
– Usually available a few days before each lecture.
– Access them from the “Lectures notes” link of the
course website (to be updated).
4
z. zheng 5
Important Course Information (2/3)
• Textbook: A. Papoulis, Probability, Random
Variables, and Stochastic Processes, McGraw-Hill
Press, 3rd Ed, 1991.
• Credits: 3 credits (48 hrs)/ 2 credits option (32 hrs,
finishes by the mid-term exam) .
• Grade Breakdown: homework 30% (50%); mid-
term 30% (50%) ; final 40%.
z. zheng 6
• Ground Rules
- No iPad, laptop and mobile phones during the
exams (therefore, don’t record the slides and
your notes only with your iPad)!
- Arrive on time, especially for sessions to be
recorded.
- Hand in your homework at the beginning of
each lecture.
Important Course Information (3/3)
z. zheng 7
Course Aims
• Understand random outcomes and random
information.
• Understand probability theory.
• Learn random variables.
• Know-how of random processes.
• Learn how to analyze statistical
information.
7
z. zheng 8
Course: Table of Contents
1. Introduction – Concept of probability and
probability spaces
– Elementary probability theory
– Conditional probability and Bayes' theorem
2. Repeated Trials – Combined experiments
– Bernoulli trials
– Poisson theorem
3. Random Variables and Distribution – Discrete and continuous random
variables
– Functions of random variables
– Joint and marginal distributions
– Independence
4. Expectation – Mean, variance and covariance
– Conditional distribution and conditional expectation
– Least squares estimation for Gaussian random vectors
5. Limit theorems – Laws of large numbers
– Central limit theorem
6. Statistics – Parameter estimation
– Hypothesis testing
z. zheng 9
Why Study Probability and Statistics?
• Decisions have impact on your life and in the world
What make decisions hard
Uncertainty (Randomness)
z. zheng 10
Why Study Probability and Statistics?
• What is Uncertainty or Randomness ?
Operator will receive a call in next one hour
z. zheng 11
Why Study Probability and Statistics?
• What is Uncertainty or Randomness ?
Variations in value of 10 resistor
Electromagnetic Interference
No. of users in network at any instant
Uncertainty is everywhere….
z. zheng 12
Why Study Probability and Statistics?
Can you give some example?
z. zheng 13
Why Study Probability and Statistics?
Can we measure uncertainty?
z. zheng 14
Why Study Probability and Statistics?
• Probability and statistics are the language of
uncertainty.
• We are able to explicitly include uncertainty
into decision making using probability.
• Statistics provide data about uncertain
relationships.
z. zheng 15
Flaws of averages: average depth is 3 feet
Can Ignore Uncertainty in Decision-Making?
Average Depth
3 ft.
z. zheng 16
Engineering Decisions Involving Uncertainty
16
Civil Engineering • Design input
- Statistics of vehicles
- Max. no. of vehicles at an instant
- Weight of vehicles
• Selection of Materials
- Analysis of Materials
• Behavior under stress
• Calculate chances of failure using
load statistics
z. zheng
Mobile Network Design
• Uncertain No. of users
• Random connection time
• Use probability theory to calculate
capacity and cell size
• Call dropping probability to ascertain
network quality
• Random channels
- Indoor
- Outdoor
- Mobile users
Telecommunications
Engineering Decisions Involving Uncertainty
17
z. zheng 18
Signal/Image Processing
Engineering Decisions Involving Uncertainty
Noise Removal
Noisy Image Clear Image
• Noise is random.
• Noise removal requires estimation of noise statistics.
• Filter is designed based on probability theory.
z. zheng 19
• Wind parameters
- Wind speed are time dependent
- wind direction
• Requires statistical approaches for
location selection and design
• Average power generation
• Requirement analysis
• Material selection
Wind Turbines
Engineering Decisions Involving Uncertainty
z. zheng 20
Industrial Engineering
• Quality of product
• Average length of life of light bulb?
• Can’t test all bulbs !!!
• Use sampling technique and use
statistical method to determine expected
life.
Engineering Decisions Involving Uncertainty
z. zheng 21
Business Management
…. and many more
• Analyzing customer demands to
determine manufacturing, marketing
and prices etc.
Engineering Decisions Involving Uncertainty
z. zheng 22
What is probability?
• One thing that makes decision hard is uncertainty.
• Probability measures uncertainty formally, quantitatively.
• Probability is the mathematical language of uncertainty.
What is probability of getting 5
at first attempt ?
22
z. zheng 23
What is Statistics?
• Statistics are numbers that summarize
the results of a study/system.
• Steps in statistics
- Collect data
- Organize or arrange the data
- Analyze the data
- Infer general conclusions
Date Jan Feb Mar Aprl
1 2 4 1 3
2 0 1 0 1
3 3 1 3 1
4 0 0 0 0
5 1 3 1 3
6 0 0 0 0
7 0 0 0 0
8 2 2 2 2
9 0 4 0 4
10 3 1 3 1
11 0 1 0 1
12 1 0 1 0
13 0 3 0 3
14 0 0 0 0
15 2 0 2 0
16 0 3 0 3
17 3 0 3 0
18 0 0 0 0
19 1 2 1 2
20 0 4 0 4
21 0 1 0 1
22 2 1 2 1
23 0 3 0 3
24 3 0 3 0
25 0 0 0 0
26 1 2 1 2
27 0 4 0 4
28 0 1 0 1
29 0 1 0 1
30 0 4 0 4
23
z. zheng 24
Link between Probability and Statistics
Model Data
Probability
Statistics
System
Prediction: looking forward
looking backward
Refine
Model
24
z. zheng 25
What is ‘probability’? (1/3)
Classical definition:
– P(A)=NA/N
– N: # of all possible outcome
NA: # of outcomes favor A
– A priori
– ‘Equal Probability’ assumption
Example:
Balls in a box.
z. zheng 26
What is ‘probability’? (2/3)
Relative frequency definition:
–
– N: # of experiments
NA: # times of A’s occurrence
– A posteriori
Example:
Toss a coin, many, many times…
n
nAP A
n lim)(
z. zheng 27
What is ‘probability’? (3/3)
Axiomatic definition:
• Experiment under repeatable conditions
• Elementary events i
• consists of i
• A, B, C, … are subsets of , denoted as
– e.g. implies
1 2Ω , , , , kζ ζ ζ
ζ A .ζ Ω
A
z. zheng 28
Set Theory--Definition
• Set: a collection of elements
• Subset: consists of elements that are also
element of the set A
• Belong or not belong to a set:
• Empty (null) set:
1 2 n , ,. . . ,A ζ ζ ζ
0
i A i A
AB
z. zheng 29
Operations on Subsets
• Union
• Intersection
• Complement
• Transitivity Property: If and ,
A B
BA
A B A
BA A
A
| or
| and
A B A B
A B A B
|A A
ζ A BA ζ B
z. zheng 30
More definitions
• Mutually exclusive (M.E) or Disjoint
B A
BA
1A2A
nA
iA
jA
• A partition of : a collection of
mutually exclusive subsets of
1
and i j i
i
A A A
z. zheng 31
De-Morgan’s Laws
A B A B
A B
BA
A B
BA
A B
BA
A B A B
A B
BA
A B
A B
A B
A B
z. zheng 32
Field
• A collection of subsets of a nonempty set
forms a field F if
• It is easy to show that etc.,
also belong to F. Thus
, , BABA
. ,,,,,,,,0 BABABABABAF
z. zheng 33
Axioms of Probability
• For any event A, the probability of A
assigned as P(A) satisfies:
i) P(A)≥0 (Probability is a nonnegative
number)
ii) P(Ω)=1 (Probability of the whole set is
unity)
iii) If A∩B=∅,then P(A∪B)=P(A)+P(B)
z. zheng 34
Some Basic Rules
•
•
• for not M.E. A and B
).(1)or P( 1)P()() P( APAAAPAA
0.P
?)( BAP
A BA
BA
, BAABA
( ) ( ) ( ) ( )P A B P A AB P A P AB
( ) ( ) ( )P B P BA P BA
( ) ( ) ( ) ( )P A B P A P B P AB
Proof:
z. zheng 35
Conditional Probability
• P(A|B) = Probability of “the event A given
that B has occurred”.
• We define provided
• This definition satisfies:
–
–
– , if
,)(
)()|(
BP
ABPBAP .0)( BP
0)|( BAP
1)|( BP
)|()|()|( BCPBAPBCAP .0CA
z. zheng 36
Properties of Conditional Probability
• If ,
• If ,
•
if and
AB ( | ) 1.P A B
BA ( | ) ( ).P A B P A
n
i
ii APABPBP1
)()|()(
ji AA
n
i
iA1
Total Probability Theorem
z. zheng 37
Bayes’ Theorem
P(A) represents the a priori probability of the event
A. Suppose B has occurred, and assume that A
and B are not independent. Bayes’ rule takes into
account the new information (“B has occurred”)
and gives out the a posteriori probability of A
given B.
( | ) ( ) ( | ) ( )P A B P B P B A P A
)()(
)|()|( AP
BP
ABPBAP
