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UNR, MATH/STAT 352, Spring 2007

Dive into the interplay between physics and probability, exploring the unpredictable nature of random phenomena and the theory of probabilities. Understand why traditional deterministic approaches fall short in dealing with complex systems with countless uncontrolled parameters.

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UNR, MATH/STAT 352, Spring 2007

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  1. Probability and Statistics MATH/STAT 352 Spring 2007 Instructor: Ilya Zaliapin zal@unr.edu UNR, MATH/STAT 352, Spring 2007

  2. 1) Physics or Probability? 2) What is Theory of Probability and what is Statistics? 3) Why do they work? UNR, MATH/STAT 352, Spring 2007

  3. Random Phenomena UNR, MATH/STAT 352, Spring 2007

  4. Degree of Predictability Predictable, certain, deterministic Unpredictable, uncertain Gravity acceleration g ~ 9.8 m/s2 A black hole h = ½ gt2 E = mgh=mv2/2 UNR, MATH/STAT 352, Spring 2007

  5. Random Phenomena Coin as a rigid body obeys classical gravitation laws, and its fall is deterministic & predictable… Coin as a gambling tool (also obeying gravitation laws) is unpredictable… HEAD? h = ½ gt2 TAIL? … it is well described by physical laws … so its fall is better described by probabilistic laws UNR, MATH/STAT 352, Spring 2007

  6. Random Phenomena Level of detail may turn deterministic phenomena into random and vice versa… Physical laws Probabilistic laws HEAD? h = ½ gt2 TAIL? UNR, MATH/STAT 352, Spring 2007

  7. Failure Examples of Random Phenomena Life time of a device (computer chip) Time Origin of randomness: micro-defects, change in temperature condition, transportation, storage, energy power,… Shooting a target with a fixed riffle Origin of randomness: defects in bullets masses, density inhomogeneities, changing atmospheric-conditions, … UNR, MATH/STAT 352, Spring 2007

  8. Physics or Probability? UNR, MATH/STAT 352, Spring 2007

  9. Physics vs. Probability Electron position measurements Max Born Angular momentum Energy Low probability High probability (December 11, 1882 – January 5, 1970) UNR, MATH/STAT 352, Spring 2007

  10. Physics vs. Probability Max Born 1954 Nobel Prize in Physics "for his fundamental research in quantum mechanics, especially for his statistical interpretation of the wavefunction" (December 11, 1882 – January 5, 1970) UNR, MATH/STAT 352, Spring 2007

  11. Physics vs. Probability Albert Einstein Quantum mechanics is certainly imposing. But an inner voice tells me it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice. (March 14, 1879 – April 18, 1955) (Letter to Max Born,1926) UNR, MATH/STAT 352, Spring 2007

  12. 2: Determine their interrelationships 3: Choose appropriate mathematical apparatus 5: Add new primary factors 4: Determine outcome given primary factors Classical (deterministic) Approach (how classical Sciences worked) 1: Single out a set of primary factors UNR, MATH/STAT 352, Spring 2007

  13. Problems of Classical Approach 20th century has witnessed the Science revolution that demonstrated a limited power of the classical approach, its inability to predict evolution of many natural phenomena. This happens because many processes crucially depend on a countless number of uncontrolled parameters and their interplay. Insignificant and undetectable changes in initial conditions may (and do) lead to different result of an entire experiment. Say, try to figure out what controls the position (head vs. tail) of a coin. UNR, MATH/STAT 352, Spring 2007

  14. What can be done? There should be a principal difference in methods and approaches applied to description of a small number of well controlled primary factors and a large number of uncontrollable secondary factors. TP & S are among sciences targeted at solving this problem (other are theory of deterministic chaos, theory of complexity, etc.) UNR, MATH/STAT 352, Spring 2007

  15. What can be done: A historic example Da Vinci (1452-1519) UNR, MATH/STAT 352, Spring 2007

  16. What can be done: A historic example Weather = Dynamics of atmosphere = Turbulence ! UNR, MATH/STAT 352, Spring 2007

  17. What can be done: a historic example Hurricane Floyd, 1999

  18. What can be done: A historic example Phenomenon Turbulence Probabilistic approach Da Vinci (1452-1519) Classical (deterministic) approach George Gabriel Stokes Andrei Kolmogorov (1903-1987): A founder of modern theory of probabilities (1933) Claude Louise Mary Henry Navier (1821) Navier-Stokes EQs UNR, MATH/STAT 352, Spring 2007

  19. What is Theory of Probability and what is Statistics? UNR, MATH/STAT 352, Spring 2007

  20. Probability: Model of Experiment Possible outcomes of coin tossing: HEAD RIB TAIL LOST Relevant, interesting, most probable outcomes: HEAD TAIL { H, T } Model: UNR, MATH/STAT 352, Spring 2007

  21. Theory of Probabilities & Statistics Probability (a fair coin will show about 50% of tails) Model Observations Statistics (a coin that shows 90 tails out of 100 throws is probably not fair) UNR, MATH/STAT 352, Spring 2007

  22. Theory of Probabilities & Statistics... … work when we can describe possible outcomes of experiment, but can not predict its specific outcome … deal with appropriate mathematical model of a physical phenomenon, not with phenomenon itself UNR, MATH/STAT 352, Spring 2007

  23. Statistical Inference sample population Riffle A hit the target with 75 bullets out of 100 Q1: What % of bullets will on average hit the target? A: 75% A: from 60% to 90% Interval estimation Point estimation Q2: How many bullets should be spent to hit the target almost surely? A: 3 bullets Q3: 14 bullets out of 100 hit the target: Was it riffle A? A: Most probably not Hypothesis testing UNR, MATH/STAT 352, Spring 2007

  24. Why do they work? UNR, MATH/STAT 352, Spring 2007

  25. Stability of Frequencies TP & S use the phenomenon of stability of frequencies: Observing a large number of uniform random events, we often detect amazing regularities # tails 1/2 % quality goods  # bets % newborn boys (about 51%) Car accidents Street traumatism Crime rates UNR, MATH/STAT 352, Spring 2007

  26. Stability of Frequencies Earthquakes: the most unpredictable Natural disaster Gutenberg-Richter law Prediction: In 2007 there will be about 1000 EQs with magnitude 5, and 10 EQs with M7 UNR, MATH/STAT 352, Spring 2007

  27. Individual vs. collective behavior $$$ = ??? $$$ = !!! UNR, MATH/STAT 352, Spring 2007

  28. Individual vs. collective behavior Luggage = ??? UNR, MATH/STAT 352, Spring 2007

  29. Conclusions TP&S do not cancel randomness, unpredictability of individual experiment. They allow to predict in some approximation an average result of a mass of uniform random phenomena. The goal of TP&S is to overpass a difficult (and often practically impossible) exploration of a single random phenomena and jump to collective behavior. Probabilistic predictions are different from exact statement of what, when, and where to expect. They establish boundary within which, with a high degree of reliability one will observe interesting phenomenon. The larger the number of individual events, the sharper the possible probabilistic prediction. UNR, MATH/STAT 352, Spring 2007

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