Probability Introduction

IEF 217a: Lecture 1Fall 2002

(no readings)

Introduction

• Probability and random variablesVery short introduction

• ParadoxesSt. PetersburgEllsberg

• Uncertainty versus risk• Computing power• Time• Chaos/complexity

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)

ii px ,

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,

N

iiii xpxE

1

)(

N

iii x

NxE

1

1)(

Variance(Dispersion)

• Expected value

• Variance,

N

iiii xpxEm

1

)(

N

iiii mxpxVar

1

2)()(

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

Evaluating a Risky Situation(Try expected value)

• Problems with E(x) or meanDispersionValuation and St. Petersburg

Dispersion

• Random variable 1Values: (4 6)Probs: (1/2, 1/2)

• Random variable 2Values: (0 10)Probs: (1/2 1/2)

• Expected ValuesRandom variable 1: 5Random variable 2: 5

Dispersion

• Possible answer:Variance

• Random variable 1Variance = (1/2)(4-5)^2+(1/2)(6-5)^2 = 1

• Random variable 2Variance = (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 payoutSum Prob(payout)*payout

1

1

1 1 1(2) (4) 8

2 4 81

( ) 22

1

i i

i

i

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

• UncertaintySome parameters (probabilities, values) not knownRisk assessments might be right or wrong

Ellsberg Paradox

• Important risk/uncertainty distinction

Ellsberg Paradox

• Urn 1 (100 balls)50 Red balls50 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 formulasMuch more complicated problems

• DisadvantageComputersOverconfidence

Two Final (difficult) Topics

• Time• Chaos/complexity

Time

• HorizonDays, weeks, months, years

• DecisionsHow effected by new information

Chaos/Complexity

• ChaosSome time series may be less random than they

appearForecasting is difficult

• ComplexityInterconnection between different variables difficult

to predict, control, or understand• Both may impact the “correctness” of our

computer models

Introduction

• Probability and random variablesVery short introduction

• ParadoxesSt. PetersburgEllsberg

• Uncertainty versus risk• Computing power• Time• Chaos/complexity