probability chapter 1

7/30/2019 Probability chapter 1

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Multimedia Network Lab.Multimedia Network Lab.

CHAPTER 1

Probability Models in Electrical andComputer Engineering

INHA UniversityINHA Universityhttp://multinet.inha.ac.krhttp://multinet.inha.ac.kr

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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http://multinet.inha.ac.kr Multimedia Network LabMultimedia Network Lab........

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

INHA UniversityINHA University 2

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%


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


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


.

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

INHA UniversityINHA University


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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.


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).


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


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


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.


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.


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=


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 =


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


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


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


probability chapter 1

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