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Physical Fluctuomatics Applied Stochastic Process 2nd Probability and its fundamental properties Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University [email protected] http://www.smapip.is.tohoku.ac.jp/~kazu/ Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 1 Probability a. Event and Probability b. Joint Probability and Conditional Probability c. Bayes Formula, Prior Probability and Posterior Probability d. Discrete Random Variable and Probability Distribution e. Continuous Random Variable and Probability Density Function f. Average, Variance and Covariance g. Uniform Distribution h. Gauss Distribution Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) This Talk Next Talk 2 Event, Sample Space and Event Experiment: Experiments in probability theory means that outcomes are not predictable in advance. However, while the outcome will not be known in advance, the set of all possible outcomes is known Sample Point: Each possible outcome in the experiments. Sample Space：The set of all the possible sample points in the experiments Event：Subset of the sample space Elementary Event：Event consisting of one sample point Empty Event：Event consisting of no sample point Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 3 Various Events Whole Events Ω:Events consisting of all sample points of the sample space. Complementary Event of Event A: Ac=Ω╲A Defference of Events A and B: A╲B Union of Events A and B: A∪B Intersection of Events A and B: A∩B Events A and B are exclusive of each other: A∩B=Ф Events A, B and C are exclusive of each other: [A∩B=Ф]Λ[B∩C=Ф]Λ[C∩A=Ф] Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 4 Empirically Definition of Probability Definition by Laplace: Let us suppose that the total number of all the sample points is N and they can occur equally Likely. Probability of an event A with N sample points is defined by p=n/N. Statistical Definition: Let us suppose that an event A occur r times when the same experiment are repeated R times. If the ratio r/R tends to a constant value p as the number of times of the experiments R go to infinity, we define the value p as probability of event A. r p R Pr A p R Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 5 Definition of Probability Definition of Kolmogorov: Probability Pr{A} for every event A in the specified sample space Ω satisfies the following three axioms: Axion 1: PrA 0 Axion 2: Pr 1 Axion 3: For every events A, B that are exclusive of each other, it is always valid that PrA B PrA PrB Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 6 Joint Probability and Conditional Probability Probability of Event A Pr{A} Joint Probability of Events A and B PrA, B PrA B Conditional Probability of Event A when Event B has happened. PrA, B PrB A PrA PrA, B PrB APrA Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) A B 7 Joint Probability and Independency of Events Events A and B are independent of each other PrA, B PrAPrB In this case, the conditional probability can be expressed as PrB A PrB A A B B Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 8 Marginal Probability Let us suppose that the sample W is expressed by Ω=A1∪A2∪…∪AM where every pair of events Ai and Aj are exclusive of each other. M PrB PrAi , B i 1 Marginal Probability of Event B for Joint Probability Pr{Ai,B} PrB PrA, B Simplified Notation A Ai B Marginalize A B Summation over all the possible events in which every pair of events are exclusive of each other. Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 9 Four Dimensional Point Probability and Marginal Probability Marginal Probability of Event B PrB PrA, B, C , D A Marginalize C D A B C D Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 10 Derivation of Bayes Formulas PrA, B PrA BPrB Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 11 Derivation of Bayes Formulas PrA, B PrA BPrB PrA, B PrB APrA Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 12 Derivation of Bayes Formulas PrA, B PrA BPrB PrA, B PrB APrA PrA, B PrA B PrB Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 13 Derivation of Bayes Formulas PrA, B PrA BPrB PrA, B PrB APrA PrA, B PrB APrA PrA B PrB PrB Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 14 Derivation of Bayes Formulas PrA, B PrA BPrB PrA, B PrB APrA PrA, B PrB APrA PrA B PrB PrB PrB A PrB APrA PrA, B PrA, B A Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 15 Derivation of Bayes Formulas PrA, B PrA BPrB PrA, B PrB APrA PrA, B PrB APrA PrA B PrB PrB PrB PrB APrA PrB APrA PrA, B PrB APrA PrA, B A A A Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 16 Derivation of Bayes Formulas PrA, B PrA BPrB A PrA, B PrB APrA B PrA, B PrB APrA PrA B PrB PrB PrB PrB APrA PrB APrA PrA, B PrB APrA PrA, B A A A Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 17 Bayes Formula PrA B PrB APrA PrB APrA Prior Probability A A Posterior Probability It is often referred to as Bayes Rule. B Bayesian Network Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 18 Summary a. Event and Probability b. Joint Probability and Conditional Probability c. Bayes Formulas, Prior Probability and Posterior Probability d. Discrete Random Variable and Probability Distribution e. Continuous Random Variable and Probability Density Function f. Average, Variance and Covariance g. Uniform Distribution h. Gauss Distribution Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) The present talk Next talk 19
