# 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%

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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