Introduction to Random Processes (1): Probability ?· 01-02-2012 · Introduction to Random Processes…

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Introduction to Random Processes (1): Probability Luiz DaSilva Professor of Telecommunications dasilval@tcd.ie +353-1-8963660 Brief review of set theory: a set q A set is a collection of elements q May be finite, or countably or uncountably infinite q Examples qA finite set: qA countably infinite set: qAn uncountably infinite set 2 Question q Consider an N-element set q How many subsets of this set are there? 3 Set algebra q Union q Intersection q Complement q Difference (reduction) 4 Set algebra example A = {odd integers between 0 and 10} B = {multiples of 3 between 1 and 10} A B = A U B = A \ B = 5 Disjoint and equal sets q Two sets are disjoint if: q Two sets are equal if: 6 De Morgans Laws 7 Random experiments qA random experiment H is an experiment for which the outcome is not known a priori qThe sample space is the set of all possible outcomes of H qAn event E is a subset of qThe null set denotes the impossible event 8 Example A random experiment: H = roll a fair die once The sample space: An event: 9 Probability Defn: Probability is a set function P[] that assigns to each event E in a -field F a number P[E] such that (i) (ii) (iii) 10 Axiom #1 P[] = 0 Proof: 11 Axiom #2 P[E] = 1 P[Ec] Proof: 12 Axiom #3 When Eis are disjoint: Proof: [A special case of property (iii) in the definition] 13 ][][11===niinii EPEPAxiom #4 P[E U F] = P[E] + P[F] P[EF] Proof: 14 Conditional probability Defn: The probability of event E conditional on event F, with P[F] > 0, is Interpretation: Provability of event E given that event F has occurred 15 ][][]|[FPEFPFEP =joint probability Example: binary communication system 16 X Y noisy channel 0 1 0 1 1 - 1 - Theorem on total probability Thm: Let E1, , En be disjoint and exhaustive (their union yields ), with P[Ei] 0. Then, for any event B, Proof: 17 ][]|[][1inii EPEBPBP ==Bayes theorem Thm: Let Eis be a partition of (disjoint, exhaustive), with P[Ei] 0. Then, for any event B, with P[B] 0, 18 === njjjiiiiEPEBPEPEBPBPBEPBEP1][]|[][]|[][][]|[Binary channel example revisited q If receiver decodes a 1, what is the probability that a 1 was sent? 19 Independence Defn: (i) Events E and F are independent if P[EF] = P[E]P[F] (ii) Events E1, , En are independent if 20 ][]...[][][][][][][][121 ====niinkjikjijijiEPEEEPkjiEPEPEPEEEPjiEPEPEEP!

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