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2600:
Probability ELEC and Random Processes in
Engineering
Spring 2012
Instructor: Prof. Jun ZHANG
Office: Rm 2448 (Lift 25-26)
Tel: 2358 7050
Email: eejzhang@ust.hk
Webpage: http://www.ece.ust.hk/~eejzhang/
1
Elec 2600: Lecture 1
• Course Details
• Models in Engineering
Their role
Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
2
Course Details
• Weekly schedule
• 2 lectures per week (Wed and Fri 4:30pm – 5:50pm, Rm 3008, Lift 3-4)
• 2 tutorials per week (Each student only needs to attend one)
• Wed 12:00pm – 12:50pm, Rm 4504, Lift 25-26
• Tue 6:00pm – 6:50pm, Rm 4504, Lift 25-26
• Grading:
• Homework (20%)
• Mid-Term (30%) + Final (50%)
• There will be 4 homework assignments
• Each worth 5%
• Details of homework submission procedures and late penalties on
website (see next slide)
Elec2600 Lecture 1
3
Course Details
• Website
• http://course.ee.ust.hk/elec2600/
• Textbook:
• Probability, Statistics and Random Processes for Electrical Engineering,
Alberto Leon-Garcia, Addison Wesley, 3rd ed., 2009.
Elec2600 Lecture 1
4
Course Details
• Course Structure
• Part I: Basic probability theory. (~2-3 weeks)
• Part II: Single random variables. (~3-4 weeks)
• Part III: Multiple random variables. (~5 weeks)
• Part IV: Stochastic processes. (~3 weeks)
•
Objective of the course
• Basic concepts of probability theory required to understand
probability models used in engineering.
• Basic techniques required to develop probability models.
Elec2600 Lecture 1
Course Details
•
Instructional Assistant
• ZHANG Xuning, eexuning@ust.hk
•
Teaching Assistants:
CHEN Shibo, schenab@ust.hk
SHI Yuanming, yshiac@ust.hk
LI Tian, tli@ust.hk
PANG Jiahao, jpang@ust.hk
•
Office location and consultation hours will be posted on website
•
TAs are there to help you – PLEASE USE THEM!
Elec2600 Lecture 1
6
Elec 2600: Lecture 1
• Course details
• Models in Engineering
Their role
Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
7
Modelling
•
•
•
•
In engineering as well as in daily life, we are faced with choices
Our decisions are based on our belief – our model – of how things will
behave
A model is an approximate representation of a physical situation which
attempts to explain observed behavior
Good models
are simple and understandable
predict all relevant aspects of the situation
•
Models can help us avoid costly (in time and money) experimentation
Elec2600 Lecture 1
8
Mathematical and Computer Simulation Models
• Mathematical model
• A set of assumptions about how a system or physical process works.
• Assumptions are stated in the form of mathematical relations between the
important parameters and variables.
• By solving these relations, we can predict the outcome of an experiment,
e.g. V = I*R
• The simpler the model, the easier it is to solve, understand and analyze.
• However, simple models may not accurately describe the actual system
of interest.
• Computer simulation model
• Assumptions are stated in the form of a computer program.
• By running the program, we can predict the outcome of an experiment,
e.g. Spice.
• Computer models can represent systems in greater detail, and more
accurately predict performance, but are generally more difficult to
analyze precisely.
Elec2600 Lecture 1
9
The Modelling Process
• Models for components
of a complex system
•
•
Validation through observation
of the physical system
•
•
Hypothesis formulation
Modification if necessary
E.g., digital camera:
•
Acquisition (sensors)
•
Digitization (quantization noise
models)
Elec2600 Lecture 1
Curve fitting,
statistical modelling, etc
10
Deterministic versus Probabilistic Models
• Deterministic Model
•
•
•
•
The conditions of the experiment determine the exact outcome
If I do A, then B will happen.
Example: Ohm’s Law (V = I*R), Kirchoff’s current and voltage laws.
In practice, there will always be random deviations if you repeat the same
experiment twice, but a deterministic model is OK if these are small.
• Probabilistic Model
• The conditions of the experiment determine the probability of different
outcomes.
• If I do A, then B will probably happen, but it is possible that C might
happen instead.
• Example: tossing of a coin, rolling a die
• In many cases, probabilistic models are used because we cannot model
all the relevant variables required by a deterministic model.
Elec2600 Lecture 1
11
More on Probabilistic Models
•
•
•
•
Allow designers to model very complex systems
Nearly all practical systems cannot be described exactly, and have some sort of
randomness
Systems can be designed and optimized “off-line”, without physical implementation –
save time and $$$
Each complex system may involve MANY models!
Elec2600 Lecture 1
12
Other Examples
Elec2600 Lecture 1
13
Elec 2600: Lecture 1
• Course details
• Models in Engineering
• Their role
• Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
14
Example: Ball Picking from an Urn
•
•
•
An urn contains 3 balls, labeled 1, 2 or 3.
A ball is selected, its number recorded (outcome) and the ball returned.
This process is repeated for n (e.g. 100) times (trials).
•
RANDOM Experiment: Outcome cannot be exactly determined!
5
4
3
2
1
0
-1
Elec2600 Lecture 1
15
Relative Frequency
• Let N1(n), N2(n) and N3(n) be the number of times that we pick
balls 1, 2, and 3 in n trials (events).
N n
• Define the relative frequency of the outcome k as f k n k
n
• This experiment exhibits
statistical regularity: as n
increases, the relative
frequency approaches a
constant value
lim f k n p k
n
where pk indicates the
probability of outcome k.
Elec2600 Lecture 1
Provides a key connection between measurement of physical
quantities and probability models!
16
Properties of the relative frequency (1)
• Since 0 ≤ Nk(n) ≤ n,
N k n
0 f k n
1
n
The relative frequencies are between zero and one.
K
• Let K = the number of possible outcomes. Since
K
k 1
f k n
K
k 1
N k n
1
n
k 1
N
k
n
n
The relative frequencies sum to one.
Elec2600 Lecture 1
17
Properties of the relative frequency (2)
• Events are groups of the outcomes of an experiment (sets):
• ONE = “the ball picked is labeled 1” = {1}
• NOT_THREE = “the ball picked is not labeled 3” = {1 or 2}
• ODD = “the ball picked is labelled with an odd number” = {1 or 3}
• We can often derive the relative frequency of one event from
the relative frequency of other events.
Example: Since NODD(n) = N1(n) + N3(n),
f ODD n
N ODD n N 1 n N 3 n
f 1 n f 3 n
n
n
• More generally, if C = {A or B} and A and B cannot occur
simultaneously exclusive),
f (mutually C n f A n f B n
Elec2600 Lecture 1
Above 3 properties coincide with AXIOMS of probability
(discussed next lecture)
18
Elec 2600: Lecture 1
• Course Details
• Models in Engineering
Their role
Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
19
Packet Voice Transmission Example
•
•
•
•
Suppose a communication system
needs to transmit N = 48
simultaneous conversations using
“packets” corresponding to 10ms
of speech.
Suppose each person speaks only
1/3 of the time.
If we wish to transmit at most M
packets every 10 ms, how should
we choose M?
Obviously, M = 48 guarantees that
no packets are ever lost. However,
this is expensive in terms of
equipment/bandwidth. In addition,
most (~2/3) of the packets will be
empty.
Elec2600 Lecture 1
On the other hand, if M < 48, some
packets may get lost.
How should we choose M so that
on average only a small fraction
(1%) of the packets are lost?
20
Example: Performance Metric
• We obtain good performance when the ratio below is small.
discard fraction =
average # of active packets lost
average # of active packets
where the average is over a large number (n) of intervals.
• Let A(j) be the number of active packets (i.e. speakers that
speak) during the jth 10ms time interval.
48
A( j )
32
16
0
0
Elec2600 Lecture 1
20
40
j
60
80
100
(n 100)
21
Example: Average number of active packets
• Define Nk(n) to be the number of intervals with k active packets.
48
15
N k (100)
A( j )
32
10
k 14
16
0
0
5
50
j
100
(n 100)
0
0
16
32
k
48
• The average number of active packets can be computed in two
different ways:
1n
1 48
average # of active packets = A( j ) k N k (n)
n j 1
n k 0
Elec2600 Lecture 1
total number of active packets
22
Example: Lost packets
• Define B(j) to be the number of packets lost in interval j.
• The number of lost packets is a function of the number of active
packets A(j).
• If A(j) ≤ M, then all packets can be transmitted and B(j) = 0.
• Otherwise, B(j) = A(j) – M.
• The maximum number of lost packets is 48 – M.
30
20
A(j)
M = 18
10
B(j)
0
0
Elec2600 Lecture 1
20
40
60
80 j 100
23
Example: Average number of lost packets
• Define Li(n) to be the number of times i packets are lost.
• As before, there are two ways to compute the average number of
lost packets
1n
1 48 M
average # of active packets lost = B( j ) i Li (n)
n j 1
n i 1
• We can also compute the average number of lost packets based
on the number of active packets.
• If the number of active packets is k and k > M, then the number
of lost packets is i = k - M. Thus,
1 48 M
# packets lost
average # of active packets lost i Li (n)
n i 1
# times k packets
appear
48
1
( k M ) N k ( n)
n k M 1
Elec2600 Lecture 1
24
Example: Express as relative frequency
• Summarizing:
1 48
1 (k M ) N k (n)
average # of active packets lost n k M
discard fraction
1 48
average # of active packets
k N k ( n)
n k 0
• Letting n → ∞,
48
48
N k ( n)
1 (k M ) n
1 (k M ) pk
k M 48
discard fraction k M 48
n
N ( n)
k k
n
k pk
k 0
k 0
0.2
• Later in the course,
we will learn how to
calculate the pk.
0.15
0.1
0.05
0
0
Elec2600 Lecture 1
Nk(100)/100
pk
16
32
k
48
25
Example (4)
• Given the pk, we can calculate the discard fraction
• Even if we transmit at most M=24 packets, on average less than
1% of the active packets will be lost.
Elec2600 Lecture 1
26
More Examples
• Casino strategy: bet $1, $2 if losing the 1st one, $4 if losing the
2nd one, and so on.
• What is the winning/losing probability?
• Communications over unreliable channels: bit “0” and bit “1”
may be interpreted incorrectly with an error probability
• Prediction of signals: use available signal portion to predict for
the future, or to analyze and synthesize signals (e.g., LPC).
Elec2600 Lecture 1
27
More Examples
• System reliability: failure probability, average time to fail
• Resource-sharing systems: banks, telephone, etc.
• Internet systems: simple client-server, peer-to-peer
• Many, many more...
Elec2600 Lecture 1
28
Elec 2600: Lecture 1
• Course Details
• Models in Engineering
• Their role
• Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
29
Axiomatic Approach to Probability
• Efforts to build a mathematical theory based on the idea of the
limit of the relative frequency run into difficulty because
•
•
It is not clear how to define the limit mathematically.
The limit cannot be found in practice.
• The axiomatic approach does not define probability as a limit,
but rather as any way of assigning numbers to events such that
they behave like the relative frequency.
• Note that the theory does not assign any meaning to the
probability, or specify how they are obtained. It only specifies
the properties it must satisfy in order for the theory to hold.
• It is up to the designer (model builder) to determine how to
assign the probability in a way that is useful.
Elec2600 Lecture 1
30
Assumptions of the Axiomatic Approach
• The set S of all outcomes has been identified.
• e.g. the number of active packets is in the set S = {0,1,2,...,48}
• A class of subsets of S called events has been specified.
• e.g. the event that number of active packets is 15
• Each event A has been assigned a number pA that satisfies
• 0 ≤ pA ≤ 1
• pS = 1
• If A and B are events that cannot occur simultaneously,
then pA or B = pA + pB
• Note the similarity with the properties of the relative frequency
covered earlier.
Elec2600 Lecture 1
31
Elec 2600: Lecture 1
• Course Details
• Models in Engineering
Their role
Types of models
•
•
•
•
Relative Frequency
Packet Voice Transmission Example
Axiomatic Approach to Probability
An Interesting Problem – Winning an iPad
Elec2600 Lecture 1
32
Something to think about (i):
• Suppose exactly 1 out of 3 boxes contains an iPad
Elec2600 Lecture 1
33
Something to think about (ii):
• You pick one randomly
Elec2600 Lecture 1
34
Something to think about (iii):
• I take away one of the boxes which you did NOT select, and I
tell you that one of the remaining boxes contains the iPad
Elec2600 Lecture 1
35
Something to think about (iv):
• I take away one of the boxes which you did NOT select, and I
tell you that one of the remaining boxes contains the iPad
Will you change your choice?
Why / Why not?
Elec2600 Lecture 1
36
Summary
•
This lecture we have discussed:
The concept of relative frequency
Practical example: Packet voice transmission
Introduction to probability axioms
37
The role and types of probability models in engineering
•
Overall course details
Example: Winning an iPad
Next Lecture: Random Experiments and Probability Axioms (more)!

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King Fahd University of Petroleum & Minerals - PETE - 305

KFUPMEdited by Foxit ReaderCopyright(C) by Foxit Corporation,2005-2009For Evaluation Only.ENGINEERINGDepartment of Petroleum EngineeringSep.28,2012PETE-305: RESERVOIR DESCRIPTIONLecture [09 and 10]course notes byDr. Enamul HossainFall 20121Re

King Fahd University of Petroleum & Minerals - PETE - 305

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King Fahd University of Petroleum & Minerals - PETE - 305

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King Fahd University of Petroleum & Minerals - PETE - 305

KFUPMEdited by Foxit ReaderCopyright(C) by Foxit Corporation,2005-2009For Evaluation Only.Oct.11,2012ENGINEERINGDepartment of Petroleum EngineeringPETE-305: RESERVOIR DESCRIPTIONLecture [13]Spatial Relationship: Estimation &Modelingcourse notes

King Fahd University of Petroleum & Minerals - PETE - 305

KFUPMENGINEERINGNov.3,2012Department of Petroleum EngineeringPETE-305: RESERVOIR DESCRIPTIONLecture [14]Modeling Variogram 1course notes byDr. Enamul HossainFall 211213.5 Modeling Variogram We have learned in Lecture 6 various methods toestim

King Fahd University of Petroleum & Minerals - PETE - 305

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King Fahd University of Petroleum & Minerals - PETE - 305

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King Fahd University of Petroleum & Minerals - PETE - 305

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King Fahd University of Petroleum & Minerals - PETE - 305

KFUPMENGINEERINGDepartment of Petroleum EngineeringPETE-305: RESERVOIR DESCRIPTIONLecture [19]Conventional Estimation Technique 4course notes byDr. Enamul HossainFall 20121Review: Ordinary Kriging KFUPM| PETE-305: Reservoir Description | Dr Ena

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King Fahd University of Petroleum & Minerals - PETE - 305

KFUPMENGINEERINGDepartment of Petroleum EngineeringPETE-305: RESERVOIR DESCRIPTIONLecture [21]Conventional Estimation Technique 6course notes byDr. Enamul HossainFall 20121AnnouncementMajor Exam IIonNov 12, 2012 @ 12:00 Noon (Monday)Venue: B

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King Fahd University of Petroleum & Minerals - PETE - 424

InstructorRock Mechanicsfor Petroleum EngineersDr. Abdulazeez AbdulraheemManaged research projects worth SR 10million in the past 10 years1Students2Students1 AL-HUSAIN, HASAN MUHAMMAD11 AL-ABDUL-MOHSEN, YOUNES A2 AL-RADHI, SALEH KHALIFAH12 AL