4.1 Mathematical Expectation

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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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4.2 Variability

• Recall our example Repair costs for a particular
  machine are represented by the following
  probability distribution
• What is the variance in the repair cost?
• That is, how might we define the spread of costs?

x 50 200 350
P(X x) 0.3 0.2 0.5
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Variance

• Remember
• µ E(X) 230
• Then, for discrete variables,
• s2 E (X - µ)2 ? (x - µ)2 f(x)
• E (X2) - µ2
• So, for our example,
• s2 ________________________

9
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 are the variance and
  standard deviation of the investors gain on the
  stock? (problem 7, pg. 94)

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Variance of Continuous Variables

• For continuous variables,
• s2 E (X - µ)2 _____________
• 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 variance of X?

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• s2(X) ? x2 f(x) dx µ2
• ________________________
• What is the standard deviation?
• s _______________
• What does this mean?

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What if the distribution is unknown?

• Chebyshevs theorem
• The probability that any random variable X will
  assume a value within k standard deviations of
  the mean is at least 1 1/k2. That is,
• P(µ ks lt X lt µ ks) 1 1/k2
• Look back at todays examples.

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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
• If g(X) 2X -1 represents the amount of
  money paid to the attendant by the manager, what
  is the variance of the amount paid on a Friday
  afternoon?
• sg(X)2 Eg(X) µg(X)2
  ____________________
• _______________________________

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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Covariance

• A measure of the nature of the association
  between two variables.
• Example 4.14