Probability Introduction

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Probability Introduction

• IEF 217a Lecture 1
• Fall 2002
• (no readings)

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Introduction

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

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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)

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Describing a Random Variable

• Histogram/picture
• Statistics
• Expected value (mean)
• Variance
• ...

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Probability Density
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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,

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Variance(Dispersion)

• Expected value
• Variance,

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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

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Evaluating a Risky Situation(Try expected value)

• Problems with E(x) or mean
• Dispersion
• Valuation and St. Petersburg

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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

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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?

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Valuation and the St. Petersburg Paradox

• Another problem for expected values

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One more probability reminder

• Compound events
• Events A and B
• Independent of each other (no effect)
• Prob(A and B) Prob(A)Prob(B)

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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)

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St. Petersburg Paradox

• Game
• Flip coin until heads occurs (n tries)
• Payout (2n) dollars
• Example
• (T T H) pays 23 8 dollars
• Prob (1/2)(1/2)(1/2)
• (T T T T H) pays 25 32 dollars
• Prob (1/2)(1/2)(1/2)(1/2)(1/2)

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What is the expected value of this game?

• Expected value of payout
• Sum Prob(payout)payout

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How much would you accept in exchange for this
game?

• 20
• 100
• 500
• 1000
• 1,000,000
• Answer none

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St. Petersburg Messages

• Must account for risk somehow
• Sensitivity to small probability events

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St. Petersburg Probability Density
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PhilosophyUncertainty 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

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Ellsberg Paradox

• Important risk/uncertainty distinction

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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

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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 dont know will go against you

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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

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Two Final (difficult) Topics

• Time
• Chaos/complexity

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Time

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

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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

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Introduction

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