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Probability Introduction

Probability Introduction. IEF 217a: Lecture 1 Fall 2002 (no readings). Introduction. Probability and random variables Very short introduction Paradoxes St. Petersburg Ellsberg Uncertainty versus risk Computing power Time Chaos/complexity. Random Variable. (Value, Probability)

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Probability Introduction

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  1. Probability Introduction IEF 217a: Lecture 1 Fall 2002 (no readings)

  2. Introduction • Probability and random variables • Very short introduction • Paradoxes • St. Petersburg • Ellsberg • Uncertainty versus risk • Computing power • Time • Chaos/complexity

  3. Random Variable • (Value, Probability) • Coin (H, T) • Prob ( ½ , ½) • Die ( 1 2 3 4 5 6 ) • Prob (1/6, 1/6, 1/6, 1/6, 1/6,1/6)

  4. Describing a Random Variable • Histogram/picture • Statistics • Expected value (mean) • Variance • ...

  5. Probability Density

  6. Expected Value (Mean/Average/Center) • Die (1/6)1+(1/6)2+(1/6)3+(1/6)4+(1/6)5+(1/6)6 • = 3.5 • Equal probability,

  7. Variance(Dispersion) • Expected value • Variance,

  8. Variance for the Die • (1/6)(1-3.5)^2 + (1/6)(2-3.5)^2 + (1/6)(3-3.5)^2+(1/6)(4-3.5)^2 + (1/6)(5-3.5)^2 + (1/6)(6-3.5)^2 • = 2.9167

  9. Evaluating a Risky Situation(Try expected value) • Problems with E(x) or mean • Dispersion • Valuation and St. Petersburg

  10. Dispersion • Random variable 1 • Values: (4 6) • Probs: (1/2, 1/2) • Random variable 2 • Values: (0 10) • Probs: (1/2 1/2) • Expected Values • Random variable 1: 5 • Random variable 2: 5

  11. Dispersion • Possible answer: • Variance • Random variable 1 • Variance = (1/2)(4-5)^2+(1/2)(6-5)^2 = 1 • Random variable 2 • Variance = (1/2)(0-5)^2+(1/2)(10-5)^2 = 25 • Is this going to work?

  12. Valuation and the St. Petersburg Paradox • Another problem for expected values

  13. One more probability reminder • Compound events • Events A and B • Independent of each other (no effect) • Prob(A and B) = Prob(A)*Prob(B)

  14. Example: Coin Flipping • Random variable (H T) • Probability (1/2 1/2) • Flip twice • Probability of flipping (H T) = (1/2)(1/2) = 1/4 • Flip three times • Prob of (H H H) = (1/2)(1/2)(1/2) = (1/8)

  15. St. Petersburg Paradox • Game: • Flip coin until heads occurs (n tries) • Payout (2^n) dollars • Example: • (T T H) pays 2^3 = 8 dollars • Prob = (1/2)(1/2)(1/2) • (T T T T H) pays 2^5 = 32 dollars • Prob = (1/2)(1/2)(1/2)(1/2)(1/2)

  16. What is the expected value of this game? • Expected value of payout • Sum Prob(payout)*payout

  17. How much would you accept in exchange for this game? • $20 • $100 • $500 • $1000 • $1,000,000 • Answer: none

  18. St. Petersburg Messages • Must account for risk somehow • Sensitivity to small probability events

  19. St. Petersburg Probability Density

  20. Philosophy:Uncertainty versus Risk(Frank Knight) • Risk • Fully quantified (die) • Know all the odds • Uncertainty • Some parameters (probabilities, values) not known • Risk assessments might be right or wrong

  21. Ellsberg Paradox • Important risk/uncertainty distinction

  22. Ellsberg Paradox • Urn 1 (100 balls) • 50 Red balls • 50 Black balls • Payout: $100 if red • Urn 2 (100 balls) • Red black in unknown numbers • Payout: $100 if red • Most people prefer urn 1

  23. What are we all doing? • People chose urn 1 to avoid “uncertainty” • Go with the cases where you truly know the probabilities (risk) • Seem to feel: • What you don’t know will go against you

  24. Computing Power and Quantifying Risk • Modern computing is creating a revolution • Move from • Pencil and paper statistics • To • Computer statistics • Advantages • No messy formulas • Much more complicated problems • Disadvantage • Computers • Overconfidence

  25. Two Final (difficult) Topics • Time • Chaos/complexity

  26. Time • Horizon • Days, weeks, months, years • Decisions • How effected by new information

  27. Chaos/Complexity • Chaos • Some time series may be less random than they appear • Forecasting is difficult • Complexity • Interconnection between different variables difficult to predict, control, or understand • Both may impact the “correctness” of our computer models

  28. Introduction • Probability and random variables • Very short introduction • Paradoxes • St. Petersburg • Ellsberg • Uncertainty versus risk • Computing power • Time • Chaos/complexity

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