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Reza Mohammadkhani, PhD
University of Kurdistan, Iran. Email: [email protected]
Stochastic Processes
Fall 2017
Outline2
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
Resources3
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 Policy4
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
Probability7
Set Definitions
Probability Space
Joint, Marginal, and Conditional Probabilities
Set Definitions8
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 ∪ 𝐴𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴
Associative Laws:𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶
Distributive Laws:𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶
BA
1A2A
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
Probability11
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 = lim𝑛→∞
𝑛𝐴𝑛
𝑛𝐴 is the number of occurrences of A
𝑛 is the total number of trials.
12
Classical Definition Relative Frequency
Conditional Probabilities13
𝑃 𝐴|𝐵 =𝑃 𝐴𝐵
𝑃 𝐵
Probability of “the event 𝐴 given that 𝐵 hasoccurred”.
Independence14
𝐴 and 𝐵 are said to be independent events, if𝑃 𝐴𝐵 = 𝑃 𝐴 𝑃 𝐵
Notes:
Two mutually exclusive events?!
Conditional probabilities?
15