# n_900 - ECE 863 Class Web Page: Welcome to ECE 863 Analysis...

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1 Welcome to ECE 863 Analysis of Stochastic Systems Part I.1: Introduction Prof. Hayder Radha Page 2 Prof. Hayder Radha ECE 863: Part I.1 ECE 863 ± Class Web Page: www.egr.msu.edu/classes/ece863 ± You need to type the whole thing in your web browser ± From Unix: /web/classes/ece863 Page 3 Prof. Hayder Radha ECE 863: Part I.1 ECE 863 ± This course teaches “mathematical tools” that are useful for a wide range of disciplines: ± Communications and Networking ± Information theory, coding, modulation, queuing theory, traffic modeling, etc. ± Signal, Speech and Image Processing ± Statistical signal processing, filtering, signal modeling, etc. ± Many other… ± Control; other areas in engineering and science Page 4 Prof. Hayder Radha ECE 863: Part I.1 ECE 863: Part I ± Introduction to Probability Theory ² Definition of random experiments ² Axioms of probability ² Mutual exclusivity ² Conditional probability ² Partition of the sample space ² Total probability ² Bay’s rule ² Independence

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2 Page 5 Prof. Hayder Radha ECE 863: Part I.1 Definition of Random Experiment Procedures/ steps (tossing a coin) Measurements/ observations Set of all possible outcomes S “Sample Space” An outcome s can NOT be decomposed into other outcomes Page 6 Prof. Hayder Radha ECE 863: Part I.1 Outcomes; events; sample space ± An event A is a set of outcomes: A = { s : such that is an even number } “outcome” Event Sample Space S B S Page 7 Prof. Hayder Radha ECE 863: Part I.1 Examples of random experiments ± Role a die once and record the result of the top-face: ± S = { 1, 2, 3, 4, 5, 6} ± = “the outcome is even” = {2, 4, 6} ± = “the outcome is larger than 3” = {4, 5, 6} ± C = “the outcome is odd” = {1, 3, 5} ± Role a die once and see if the top-face is even ± S = { even, odd } = { , C } Page 8 Prof. Hayder Radha ECE 863: Part I.1 Axioms of Probability ± Probability of any event is non-negative: P[ ] 0 ± The probability that “the outcome belongs to the sample space” is 1: P[S] = 1 ± The probability of “the union of mutually- exclusive events” is the sum of their probabilities: If 1 2 = , P[ 1 2 ]=±P A 1 ] + A 2
3 Page 9 Prof. Hayder Radha ECE 863: Part I.1 Mutual Exclusivity ± The probability of “the union of mutually- exclusive events” is the sum of their probabilities: If A i A j = , i j = j j j j A P A P ] [ ± A 2 S 1 3 Page 10 Prof. Hayder Radha ECE 863: Part I.1 Mutual Exclusivity ± However , in general: P[ 1 2 ] = P[ 1 ] + P[ 2 ] - P[ 1 2 ] This formula works for both mutually exclusive and non-mutually-exclusive events 2 S 1 3 Page 11 Prof. Hayder Radha ECE 863: Part I.1 Example I.1 ± Role a die twice and record the number of dots on the top-face: S = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5)

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## This note was uploaded on 03/26/2012 for the course DTVT 302 taught by Professor Tuan during the Spring '12 term at Mt. Vernon Nazarene.

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n_900 - ECE 863 Class Web Page: Welcome to ECE 863 Analysis...

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