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) Coin (H, T) Prob ( ½, ½) Die ( ) Prob (1/6, 1/6, 1/6, 1/6, 1/6,1/6)

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

Probability Density

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,

Variance (Dispersion) Expected value Variance,

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 =

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

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

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?

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

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

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)

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)

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

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

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

St. Petersburg Probability Density

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

Ellsberg Paradox Important risk/uncertainty distinction

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

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

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

Two Final (difficult) Topics Time Chaos/complexity

Time Horizon + Days, weeks, months, years Decisions + How effected by new information

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

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