probability, random processes and inference - cic pescamilla/prpi/slides/prpi_1.pdf ·...

Download Probability, Random Processes and Inference - CIC pescamilla/PRPI/slides/PRPI_1.pdf · Probability,…

Post on 30-Aug-2018




0 download

Embed Size (px)



    Probability, Random Processes and Inference

    Dr. Ponciano Jorge Escamilla

    Laboratorio de


  • CIC


    Dr. Ponciano Jorge Escamilla Ambrosio

    Class meetings

    Mondays and Wednesdays 12:00 14:00 hrs.

    Classroom Aula A3


    Probability, Random

    Processes and Inference

  • CIC

    Course web site:

    Reader material and homework exercises, etc.


    Course web site

  • CIC

    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.


    Course Objective

  • CIC


    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

  • CIC


    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

  • CIC


    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

  • CIC


    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


    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

  • CIC


    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

  • CIC


    Course text books

    Joseph Blitzstein, Jessica Hwang. Introduction to probability, CRC Press2014.

    Dimitri P. Bertsekas and John N. Tsitsiklis. Introduction to probability, 2nd Edition, Athena Scientific, 2008.

  • CIC


    Course text books

    Gza Schay, Introduction to probability with statistical applications,Birkhauser, Boston, 2007.

    William Feller. An introduction to probability theory and its applications, Vol. 1, 3rd Edition, Wiley, 1968.

  • CIC

    Midterm exam 15%

    Final exam 15%

    Homework assignments 20%

    One written departmental exam 50%



  • CIC


    Course Schedule A-17

  • CIC

    1. What is Probability?

    1.1.1. Statistical Probability

    1.1.2. Probability as a Measure of Uncertainty



  • CIC


    What is Probability?

  • CIC

    The relative is trying to use the concept of

    probability to discuss an uncertain


    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


    What is Probability?

  • CIC

    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?



    What is Probability?

  • CIC

    Example: Flip a coin twice


    What is Probability?



  • CIC

    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.


    Sample space

  • CIC

    Tossing twice a coin experiment, example


    Sample space

  • CIC

    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]


    What is Probability?

  • CIC

    Provides tools for understanding and

    explaining variation, separating signal from

    noise, and modeling complex phenomena.

    (engineer definition)


    What is Probability?

  • CIC

    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


    What is Probability?

  • CIC

    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:



    What is Probability?

  • CIC


    Probabilistic Model

  • CIC

    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



    Elements of a Probabilistic Model

  • CIC

    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.


    Experiments and events

  • CIC

    There is no restriction on what constitutes an


    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.


    Experiments and events

  • CIC

    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.


    Experiments and events

  • CIC

    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


View more >