Graduate School of Information Sciences, Tohoku University Physical Fluctuomatics Applied Stochastic Process 2nd Probability and its fundamental properties Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) Probability Event and Probability Joint Probability and Conditional Probability Bayes Formula, Prior Probability and Posterior Probability Discrete Random Variable and Probability Distribution Continuous Random Variable and Probability Density Function Average, Variance and Covariance Uniform Distribution Gauss Distribution This Talk Next Talk Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

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)

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 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)

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. Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

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: Axion 2: Axion 3: For every events A, B that are exclusive of each other, it is always valid that Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Joint Probability and Conditional Probability Probability of Event A Joint Probability of Events A and B Conditional Probability of Event A when Event B has happened. A B Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Joint Probability and Independency of Events Events A and B are independent of each other In this case, the conditional probability can be expressed as A B A B Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) Marginal Probability Let us suppose that the sample space W is expressed by Ω=A1∪A2∪…∪AM where every pair of events Ai and Aj is exclusive of each other. Ai B Marginal Probability of Event B for Joint Probability Pr{Ai,B} Marginalize A B Simplified Notation Summation over all the possible events in which every pair of events are exclusive of each other. Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Four Dimensional Point Probability and Marginal Probability Marginal Probability of Event B A B C D Marginalize Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Derivation of Bayes Formulas Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) Bayes Formula Prior Probability A B Posterior Probability It is often referred to as Bayes Rule. Bayesian Network Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) Summary Event and Probability Joint Probability and Conditional Probability Bayes Formulas, Prior Probability and Posterior Probability Discrete Random Variable and Probability Distribution Continuous Random Variable and Probability Density Function Average, Variance and Covariance Uniform Distribution Gauss Distribution The present talk Next talk Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)