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INSTITUTO POLITCNICO NACIONALCENTRO DE INVESTIGACION EN COMPUTACION

Probability, Random Processes and Inference

Dr. Ponciano Jorge Escamilla Ambrosiopescamilla@cic.ipn.mx

http://www.cic.ipn.mx/~pescamilla/

Laboratorio de

Ciberseguridad

CIC

Instructor

Dr. Ponciano Jorge Escamilla Ambrosio

pescamilla@cic.ipn.mx

http://www.cic.ipn.mx/~pescamilla/

Class meetings

Mondays and Wednesdays 12:00 14:00 hrs.

Classroom Aula A3

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Probability, Random

Processes and Inference

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Course web site:

http://www.cic.ipn.mx/~pescamilla/academy.html

Reader material and homework exercises, etc.

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Course web site

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The student will learn the fundamentals of

probability theory: probabilistic models, discrete

and continuous random variables, multiple

random variables and limit theorems as well as an

introduction to more advanced topics such as

random processes and statistical inference. At the

end of the course the student will be able to

develop and analyse probabilistic models in a

manner that combines intuitive understanding and

mathematical precision.

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Course Objective

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Course content

1. Probability

1.1. What is Probability?

1.1.1. Statistical Probability

1.1.2. Probability as a Measure of Uncertainty

1.2. Sample Space and Probability

1.2.1. Probabilistic Models

1.2.2. Conditional Probability

1.2.3. Total Probability Theorem and Bayes Rule

1.2.4. Independence

1.2.5. Counting

1.2.6. The probabilistic Method

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Course content

1.3. Discrete Random Variables

1.3.1. Basic Concepts

1.3.2. Probability Mass Functions

1.3.3. Functions of Random Variables

1.3.4. Expectation and Variance

1.3.5. Joint PMFs of Multiple Random Variables

1.3.6. Conditioning

1.3.7. Independence

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Course content

1.4. General Random Variables

1.4.1. Continuous Random Variables and PDFs

1.4.2. Cumulative Distribution Function

1.4.3. Normal Random Variables

1.4.4. Joint PDFs of Multiple Random Variables

1.4.5. Conditioning

1.4.6. The Continuous Bayes Rule

1.4.7. The Strong Law of Large Numbers

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Course content

2. Introduction to Random Processes

2.1. Markov Chains

2.1.1. Discrete Time Markov Chains

2.1.2. Classification of States

2.1.3. Steady State Behavior

2.1.4. Absorption Probabilities and Expected Time to

Absorption

2.1.5. Continuous Time Markov Chains

2.1.6. Ergodic Theorem for Discrete Markov Chains

2.1.7. Markov Chain Montecarlo Method

2.1.8.Queuing Theory

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Course content

3. Statistics

3.2. Classical Statistical Inference

3.2.1. Classical Parameter Estimation

3.2.2. Linear Regression

3.2.3. Analysis of Variance and Regression

3.2.4. Binary Hypothesis Testing

3.2.5. Significance Testing

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Course text books

Joseph Blitzstein, Jessica Hwang. Introduction to probability, CRC Press2014.https://www.crcpress.com/Introduction-to-Probability/Blitzstein-Hwang/9781466575578

Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability, 2nd Edition, Athena Scientific, 2008. http://athenasc.com/probbook.html

https://www.crcpress.com/Introduction-to-Probability/Blitzstein-Hwang/9781466575578http://athenasc.com/probbook.html

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Course text books

Gza Schay, Introduction to probability with statistical applications,Birkhauser, Boston, 2007.http://link.springer.com/book/10.1007/978-0-8176-4591-5

William Feller. An introduction to probability theory and its applications, Vol. 1, 3rd Edition, Wiley, 1968.http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471257087.html

http://link.springer.com/book/10.1007/978-0-8176-4591-5http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471257087.html

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Midterm exam 15%

Final exam 15%

Homework assignments 20%

One written departmental exam 50%

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Grading

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Course Schedule A-17http://www.cic.ipn.mx/~pescamilla/academy.html

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1. What is Probability?

1.1.1. Statistical Probability

1.1.2. Probability as a Measure of Uncertainty

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Probability

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What is Probability?

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The relative is trying to use the concept of

probability to discuss an uncertain

situation

Luck, Coincidence, Randomness,

Uncertainty, Risk, Doubt, Fortune, Chance

Used in a vague, casual way!

A first approach to define probability is in

terms of frequency of occurrence, as a

percentage of success

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What is Probability?

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For example, if we toss a coin, and observe

whether it lands head (H) or tail (T) up

What is the probability of either result?

Why?

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What is Probability?

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Example: Flip a coin twice

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What is Probability?

=#

#

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Definition 1 (Sample space and event).

The sample space S of an experiment is the

set of all possible outcomes of an experiment.

An event A is a subset of the sample space S,

and we say that A occurred if the actual

outcome is in A.

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Sample space

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Tossing twice a coin experiment, example

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Sample space

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Probability is logical framework for

quantifying uncertainty and randomness [Blitzstein and Hwang, 2014]

Probability theory is a branch of

mathematics that deals with repetitive events

whose occurrence or nonoccurrence is

subject to chance variation. [Schay, 2007]

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What is Probability?

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Provides tools for understanding and

explaining variation, separating signal from

noise, and modeling complex phenomena.

(engineer definition)

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What is Probability?

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There are situation where the frequency

interpretation is not appropriate

Example: A scholar asserts that the Iliad and

the Odyssey were composed by the same

person, with probability 90%

It is based on the scholars subjective belief

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What is Probability?

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The theory of probability is useful in a broad

variety of contexts and applications:

Statistics, Physics, Biology, Computer

Science, Meteorology, Gambling, Finance,

Political Science, Medicine, Life.

Assignment 1a: Give an example of the

application of probability theory in each area

Assignment 1b: Read math review: http://projects.iq.harvard.edu/files/stat110/files/math_rev

iew_handout.pdf

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What is Probability?

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Probabilistic Model

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The sample space S, which is the set of all

possible outcomes of an experiment.

The probability law, which assigns to a set A of

possible outcomes (also called an event) a

nonnegative number P(A) (called the probability

of A) that encodes our knowledge or belief about

the collective likelihood of the elements of A.

The probability law must satisfy certain

properties.

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Elements of a Probabilistic Model

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The experiment will produce exactly one out of

several possible outcomes.

A subset of the sample space, that is, a collection

of possible outcomes, is called an event.

It means that any collection of possible

outcomes, including the entire sample space S

and its complement, the empty set , may

qualify as an event.Strictly speaking, however, some sets have to be excluded. In particular when dealing with probabilistic models

involving an uncountable infinite sample space, there are certain unusual subsets for which one cannot

associate meaningful probabilities.

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Experiments and events

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There is no restriction on what constitutes an

experiment.

The events to be considered can be described by

such statements as a toss of a given coin results

in head, a card drawn at random from a regular

52 card deck is an Ace, or this book is green.

Associated with each statement there is a set S of

possibilities, or possible outcomes.

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Experiments and events

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Examples of experiments and events:

Tossing a Coin. For a coin toss, S may be taken to consist of

two possible outcomes, which we may abbreviate as H and T

for head and tail. We say that H and T are the members,

elements or points of S, and write S = {H, T}.

Tossing two coins but ignore one of them. In this case S =

{HH, HT, TH, TT}. In this case, for instance, the outcome

the first coin shows H is represented by the set {HH, HT},

that is, this statement is true if we obtain HH or HT and false

if we obtain TH or TT.

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Experiments and events

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Tossing a Coin Until an H is Obtained. If we toss a coin

until an H is obtained, we cannot say in advance how many

tosses will be required, and so the natural sample space is S =

{H, TH, TTH, TTTH, . . . }, an infinite set. We can use, of

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