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L1plus_probty_overview_PartA_v1 - ProbabilityAn Overview B...

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Probability–An Overview B. Sainath EEE Dept., BITS PILANI Aug, 2015 B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 1 / 51
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Outline 1 Why Should We Study Probability? 2 Terminology & Basic Axioms and Rules 3 Conditional Probability, Bayes’ Rule & Examples 4 Random Variables 5 Random Processes 6 Further Reading & References B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 2 / 51
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Why Should We Study Probability Mathematical modeling and analysis of random experiments Helps us to understand statistical regularity Enable to predict Applications Communication and signal processing (eg. in detection, estimation) & Networking (eg. Markov chains) Stochastic finance Quantum mechanics ..so on. B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 3 / 51
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Terminology & Basic Axioms (Due to Kolmogorov) Terminology Random experiment Outcomes are not known a priori Eg. Tossing a coin, Throwing a die, Measurement of received signal embedded in noise, so on. Sample space Ω (universal set) Set of all sample points ( ω s), each representing one outcome of experiment Types: Continuous & Discrete Outcomes are disjoint σ -field or algebra F Of sets of points of Ω Sets in σ –field are events Which is sure/certain event ? (ans: Set Ω ) Which is impossible event? (ans: Set φ ) Probability measure P Provides probability for every set in F Countably additive measure defined on F P (Ω) = 1 and P ( φ ) = 0 B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 4 / 51
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Terminology & Basic Axioms Probability space: a triplet ( Ω , F , P ) Axioms (satisfied by σ –algebra): 1 Ω ∈ F 2 If A ∈ F then A c ∈ F 3 If A ∈ F , B ∈ F then A B ∈ F 4 A 1 , A 2 , . . . , are sequence of events in F ⇒ S i = i = 0 A i ∈ F Show the following: A B ∈ F T i = i = 0 A i ∈ F (hint: use axioms) Mutually exclusive events (MEE): A B = φ B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 5 / 51
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More Axioms Axioms satisfied by P 1 P ≥ 0 for all A ∈ F 2 A , B mutually exclusive: P ( A B ) = P ( A ) + P ( B ) 3 Sequence of MEEs: P ( i = [ i = 0 A i ) = P ( i = X i = 0 A i ) 4 P (Ω) = 1 Figure: MEEs–Venn diagram illustration image source: http://www.drcruzan.com/MathProbability.html B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 6 / 51
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Various Events Complementary event: A c Union: A B Intersection: A B Subset: A B Equality: A = B iff A B and B A MEE: A B = φ Partition: Figure: Set partitioning–Venn diagram illustration. Source: http : // www . projectrhea . org / rhea / index . php / From Bayes Theorem to Pattern Recognition via Bayes Rule B. Sainath (BITS, PILANI) Probability–An Overview Aug, 2015 7 / 51
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Some Rules of Combining Events For any subsets A, B, C of F ( A c ) c = A , A A c = Ω , A A c = φ Commutative A B = B A A B = B A Associative A ( B C ) = ( A B ) C A ( B C ) = ( A B ) C Distributive A ( B C ) = ( A B ) ( A C ) A ( B C ) = ( A B ) ( A C ) Demorgan’s laws ( A B ) c = A c B c ( A B ) c = A c B c Show the following using axioms. Assume probability space discussed.
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