probability introduction ief 217a: lecture 1 fall 2002 (no readings)
Post on 22-Dec-2015
214 views
TRANSCRIPT
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 ,
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 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?
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
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
• 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
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