probability chapter 1
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CHAPTER 1
Probability Models in Electrical andComputer Engineering
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Prof. SangProf. SangProf. SangProf. Sang----Jo YooJo YooJo YooJo Yoo
[email protected]@[email protected]@inha.ac.kr
http://multinet.inha.ac.krhttp://multinet.inha.ac.krhttp://multinet.inha.ac.krhttp://multinet.inha.ac.kr
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Course Info. Course objective
To learn basics of probability and random processes
Text Book A. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical
Engineering, 3rd Ed.
Instructor
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Prof. Sang-Jo Yoo (Office: Hi-Tech 316, Tel: 860-8304, E-mail: [email protected]) Office hours: by appointment (Telephone or E-mail)
Course Home Page: http://multinet.inha.ac.kr/
Homework: Assignment are issued weekly and due at the beginning of class one week after
they are assigned
Late/copied assignments will never be accepted.
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Course Info.
Exams:
2 exams1 mid-term and 1 final exam
Closed book
No make-ups for missed exams (Any request for exceptions should be made inadvance)
Grading:
Attendant: 10%
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Homework: 10% Mid-term: 40%
Final exam: 40%
e-Class:
Lecture notes and homework will be posted on e-Class
Encourage the use of Q & A board on e-Class web site
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Course Objectives In this course we will study basics of probability and random
processes:
Probability models in electrical engineering
Basic concepts of probability theory
Random variables
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Multiple random variables Random processes
Analysis and processing of random signals
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Models in Analysis and Design Need to build systems that work in a chaotic environment
Communication systems that must provide continuous and error-freecommunication.
Computer system that must satisfy the diverse processing demands.
Models as tools in analysis and design
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.
Actual implementation and experimental evaluation of the alternatives aretoo costly.
Models can be used to make decisions.
Probability models enable the designers to successfully build systems that
are efficient, reliable, and cost-effective.
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Models in Analysis and Design
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Mathematical Models Model
An approximate representation of a physical system.
Attempts to explain observed behavior using a set of simple andunderstandable rules.
Mathematical models
Used when the observational phenomenon has measurable properties.
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Consists of a set of assumption about how a system works. Stated in the form of mathematical relations involving parameters and
variable of the system.
How we can validate the mathematical model many real observations?
Computer simulation model Consists of computer program that simulates the system.
A means of checking the predictions made by a mathematical model.
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Types of Models Deterministic model
The conditions under which an experiment is carried out determine theexact outcome of the experiment.
The solution of a set of mathematical equations specifies the exactoutcome of the experiment (e.g., circuit theory).
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ro a ty mo e For systems involving unpredictable variation and randomness.
Random experiment: an experiment in which the outcome varies in anunpredictable fashion when the experiment is repeated under the sameconditions.
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Example: Electrical Network
1. Represent the given circuit usingsystems of linear equations
2. Find the solution
1 3I
2I
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VRI
I
I
I
IIIIIIII
IIIII
IIIII
=
=
=+=++
==+
==++
0
6
6
2136
3120
6010
021360)(312)(6
631206)(39
661006)(64
3
2
1
32123313
32322
31311
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Probability Model
Random experiment
An experiment where the outcome varies in an unpredictable fashionwhen the experiment is repeated under the same conditions.
(i) It is performed according to some set of rules,
(ii) It can be repeated arbitrarily often,
(iii) The result of each performance depends on chance and cannot be
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predicted uniquely. Sequential random experiments performing a sequence of simple
random sub-experiments E.g., First toss a coin, then throw a dice.
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Probability Model Example of a random experiment
A ball is selected from an urn containing three identical balls,
labeled 0, 1, and 2
Sample space S:The set of all possible outcomes In this example, S={0, 1, 2}
The outcome of the experiment is random.
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0 10 20 30 40 50 60 70 80 90 100-2
-1
0
1
2
3
Trial number
Outcome
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Statistical Regularity A model must enable us to make predictions about the future
behavior of a system.
Statistical regularity Average obtained in long sequences of repetitions of random
experiments yield approximately the same value.
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Example Suppose the urn experiment is repeated ntimes; k=0,1,2
Relative frequency
Probability of the outcome kn
nNnf
kk
)()( =
kk
n
Pnf =
)(lim
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Example: Relative Frequency
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Properties of Relative Frequency The number of occurrences of an outcome
The relative frequencies
0 ( ) 1kf n
0 ( )kN n n
for 1, 2,...,k K=
for 1, 2,...,k K=
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The sum of all the relative frequencies
Event C = Aor Boccurs (cannot occur simultaneously)
==
==
KK
11
)(1)(k
k
k
k nnNnf Q
( ) ( ) ( )c a bf n f n f n= +
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Axioms of ProbabilityRelative frequency: problems
Not clear when and it what mathematical sense the limit exist.
We cannot perform an experiment an infinite number of times.
Probability of an event
1. 0 ( ) 1P A 1S =
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3. IfAand Bare events that cannot occur simultaneously,
then [ or ] [ ] [ ]P A B P A P B= +
S
A B
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Building A Probability Model Procedures for building a probability model
Define the random experiment inherent in the application
Specify the sample space Sof all possible outcomes and events ofinterest.
Specify a probability assignment from which the probability of all eventsof interest can be computed
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Example
Test a telephone conversation to determine whether a speaker iscurrently speaking or not.
On the average, Pactive
=1/3, Psilent
=2/3
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Example: Telephone Conversation
On the average, Pactive=1/3, Psilent=2/3
Physical situation
What is the probability that more than 24 speakers out of 48 independent speakersare active at the same time?
Can we use a probability model?
Assumptions
All speakers are the same
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All languages have the same silence-activity behavior
Build a probability model
Use an urn experiment
Active period black ball, silent period white ball
Select a ball from an urn containing two white balls and one black ball
(Pactive=1/3, Psilent=2/3) What is the probability that more than 24 black balls are selected in 48
independent repetitions of the urn experiment?
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Example: Probability Model Binary communication channel
Every Tseconds, the transmitter accepts a binary input (0 or 1) andtransmits a signal
The receiver makes a decision for the receiver signal
Transmitter errors may occur with probability
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