4.1%20Mathematical%20Expectation

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4.1 Mathematical Expectation

• Example Repair costs for a particular machine
  are represented by the following probability
  distribution
• What is the expected value of the repairs?
• That is, over time what do we expect repairs to
  cost on average?

x 50 200 350
P(X x) 0.3 0.2 0.5
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Expected value

• µ E(X)
• µ mean of the probability distribution
• For discrete variables,
• µ E(X) ? x f(x)
• So, for our example,
• E(X) ________________________

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Your turn

• By investing in a particular stock, a person can
  take a profit in a given year of 4000 with a
  probability of 0.3 or take a loss of 1000 with a
  probability of 0.7. What is the investors
  expected gain on the stock? (problem 7, pg. 94)

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Expected value of Continuous Variables

• For continuous variables,
• µ E(X) _______
• Example Recall from last time, problem 7 (pg.
  73)
• x, 0 lt x lt 1
• f(x) 2-x, 1 x lt 2
• 0, elsewhere
• (in hundreds of hours.)
• What is the expected value of X?

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• E(X) ? x f(x) dx
• ________________________

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Functions of Random Variables

• Example 4.4. Probability of X, the number of cars
  passing through a car wash in one hour on a sunny
  Friday afternoon, is given by
• Let g(X) 2X -1 represent the amount of
  money paid to the attendant by the manager. What
  can the attendant expect to earn during this hour
  on any given sunny Friday afternoon?
• Eg(X) S g(x) f(x)
  ____________________
• _______________________________

x 4 5 6 7 8 9
P(X x) 1/12 1/12 1/4 1/4 1/6 1/6
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Your Turn

• Problem 20, pg. 95