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  • Reza Mohammadkhani, PhD

    University of Kurdistan, Iran. Email: Mohammadkhani@gmail.com

    Stochastic Processes

    Fall 2017

  • Outline 2

     Probability and Random Variables  Probability and Random Variables

     Distribution Functions

     Joint, Marginal and Conditional Probability Functions

     Functions of Random Variables

     Statistical Averages (Expected Values)

     Simulations by MATLAB

     Stochastic Processes  Classifications (Stationarity, Ergodicity, etc.)

     Correlation Functions

     Power Spectrum

     Simulations by MATLAB

     Applications:  Detection and Estimation Theory

     Filtering and Prediction

  • Resources 3

    Required:  Lecture notes

     A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes, 4th Edition, McGraw Hill, 2002.

    Recommended books:  J. G. Proakis, M. Salehi, and G. Bauch, Contemporary Communication Systems

    Using MATLAB, 3rd edition, Cengage Learning, 2012.

     K. Sam Shanmugan, Arthur M. Breipohl, Random Signals: Detection, Estimation and Data Analysis, Wiley, 1988.

     A. Leon-Garcia, Probability and Random Processes for Electrical Engineering, 3rd Edition, Prentice Hall, 2008.

     M. Barkat, Signal Detection and Estimation, 2nd edition, Artech House, 2005.

  • Grading Policy 4

     Midterm exam: 35%

     Final exam: 35%

     Homeworks/Projects: 20%

     Attendance/Quizzes: 10%

    Course Webpage: http://eng.uok.ac.ir/mohammadkhani/courses/StochasticProcesses.html

  • 5

  • Review of Probability6

  • Probability 7

     Set Definitions

     Probability Space

     Joint, Marginal, and Conditional Probabilities

  • Set Definitions 8

     Null/empty set: ∅

     Whole/entire set: 𝑆

     Union: 𝐴 ∪ 𝐵

     Intersection: 𝐴 ∩ 𝐵

     Complement: ҧ𝐴

    A B

    𝐴 ∪ 𝐵 𝐴 ∩ 𝐵

    A B A

    ҧ𝐴

    ҧ𝐴

  • 9

     Mutually Exclusive sets:

     For two arbitrary sets 𝐴 and 𝐵: 𝐴 ∩ 𝐵 = ∅

     For 𝑛 sets of 𝐴1, 𝐴2, … , 𝐴𝑛 : 𝐴𝑖 ∩ 𝐴𝑗 = ∅ for each 𝑖 ≠ 𝑗

     Commutative Laws: 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴

     Associative Laws: 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶 𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶

     Distributive Laws: 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶

    BA

    1A 2A

    nA

    iA

    jA

  • 10

     DeMorgan’s Laws:

    𝐴 ∩ 𝐵 = ҧ𝐴 ∪ ത𝐵

    𝐴 ∪ 𝐵 = ҧ𝐴 ∩ ത𝐵

    A B A B A B

    𝐴 ∪ 𝐵 𝐴 ∪ 𝐵 ҧ𝐴 ∩ ത𝐵

    BA

  • Probability 11

     Random experiment:

    its outcome is not known in advance

     Sample Space:

    All possible outcomes of a random experiment

     Random event

    𝑃 𝐴 : Probability of an event 𝐴

  • Probability

     Probability of an event 𝐴

    𝑃 𝐴 = 𝑛 𝐴

    𝑛 𝑆

     Probability of an event 𝐴

    𝑃 𝐴 = lim 𝑛→∞

    𝑛𝐴 𝑛

     𝑛𝐴 is the number of occurrences of A

     𝑛 is the total number of trials.

    12

    Classical Definition Relative Frequency

  • Conditional Probabilities 13

    𝑃 𝐴|𝐵 = 𝑃 𝐴𝐵

    𝑃 𝐵

     Probability of “the event 𝐴 given that 𝐵 has occurred”.

  • Independence 14

     𝐴 and 𝐵 are said to be independent events, if 𝑃 𝐴𝐵 = 𝑃 𝐴 𝑃 𝐵

     Notes:

     Two mutually exclusive events?!

     Conditional probabilities?

  • 15

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