# stochastic processes - uok.ac.ir outline 2 probability and random variables probability and random...

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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%

• 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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