Expected Value

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Expected Value
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Expected Value

• In gambling on an uncertain future, knowing the
  odds is only part of the story!
• Example I flip a fair coin.
• If it lands HEADS, you get .
• If it lands TAILS, you give me 1.
• Do you take the bet?

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Expected Value

• All these scenarios carry the same sample space
  and same probabilities
• S H, T , P(H) P(T) ½
• But they do NOT all carry the same monetary
  outcome!

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Expected Value

• Knowing the odds is only half the battle, if
  different outcomes have different value to you.
• Random Variable
• A random variable is a function X which assigns a
  numerical value to each outcome in a sample
  space.
• Example If we decide to workout John Sanders
  Loan, the sample space is
• S success, failure
• Each outcome has a different monetary value to
  Acadia Bank.

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Expected Value

• Let X be the amount of money we get back from a
  loan workout
• X 4,000,000 or X 250,000
• We are assigning a numerical value to each event
  in the sample space.
• Were describing each event with a number!

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Expected Value

• Describing events with a random variable
• Let X sum obtained from a roll of two dice.
• Let E be the event sum of the dice is greater
  than 8
• Let F be the event the sum of the dice is between
  3 and 6.
• How do we describe the events E and F using our
  random variable X?
• E could be described as P(X gt 8)
• F could be described as P(3 X 6)

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Expected Value

• What is P(X gt8) ?
• What is P(3 X 6) ?

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Expected Value

• Assigning a random variable to our probability
  space helps us balance risk with reward.
• Fair-coin flipping example
• Let X be our net profit from the game

Reward exceeds risk. We should play!
Risk equals reward. The game is fair
Risk exceeds reward. We shouldnt play!
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Expected Value

• What happens when the events are not equally
  probable?
• S HEADS, TAILS and P(H) 0.25, P(T) 0.75
• What should the payoffs X be for this game to be
  fair?
• So you should receive 3 for heads if you have to
  pay 1 for tails.

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Expected Value

• If we play this game 1000 times, how much can we
  expect to win or lose?
• If we play a 1000 times
• 250 heads, at 3 a piece means we receive 750
• 750 tails, means we lose 1 so we pay -750
• After 1000 tosses, we net 0
• This net is known as the expected value of the
  random variable X.

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Expected Value

• The expected value of any random variable is the
  average value we would expect it to have over a
  large number of experiments.
• It is computed just like on the previous slide
• for n distinct outcomes in an experiment!

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Expected Value

• Note that the notation asks for the
  probability that the random variable represented
  by X is equal to a value represented by x.
• Remember that for n distinct outcomes for X,
• (The sum of all probabilities equals 1).

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Expected Value

• Ex. Consider tossing a coin 4 times. Let X be
  the number of heads. Find and
  .

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Expected Value

• Soln.

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Expected Value

• Ex. Find the expected value of X where X is the
  number of heads you get from 4 tosses. Assume
  the probability of getting heads is 0.5.
• Soln. First determine the possible outcomes.
  Then determine the probability of each. Next,
  take each value and multiply it by its
  respective probability. Finally, add these
  products.

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Expected Value

• Possible outcomes
• 0, 1, 2, 3, or 4 heads
• Probability of each

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Expected Value

• Take each value and multiply it by its
  respective probability
• Add these products
• 0 0.25 0.75 0.75 0.25 2

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Expected Value

• Ex. A state run monthly lottery can sell 100,000
  tickets at 2 apiece. A ticket wins 1,000,000
  with probability 0.0000005, 100 with probability
  0.008, and 10 with probability 0.01. On
  average, how much can the state expect to profit
  from the lottery per month?

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Expected Value

• Soln. States point of view
• Earn Pay Net
• 2 1,000,000
  -999,998
• 2 100 -98
• 2 10 -8
• 2 0 2
• These are the possible values. Now find
  probabilities

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Expected Value

• Soln. States point of view
• We get the last probability since the sum of all
  probabilities must add to 1.

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Expected Value

• Soln. States point of view
• Finally, add the products of the values and their
  probabilities

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Expected Value

• Focus on the Project
• X amount of money from a loan work out
• Compute the expected value for typical loan

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Expected Value

• Focus on the Project
• What does this tell us?
• Foreclosure 2,100,000
• Ave. loan work out 1,991,000
• Tentatively, we should foreclose. This doesnt
  account for the specific characteristics of J.
  Sanders. However, this could reinforce or weaken
  our decision.