1. 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
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

2. 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)

3. 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)

4. 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)

5. 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)

6. 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)

7. 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)

8. 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)

9. 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)

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

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

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

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

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

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

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

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

18. 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)

19. 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)