Lecture 10: Tue, Oct 7

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Lecture 10 Tue, Oct 7

• Announcements
• HW 4 due Thu
• Exam on Oct 14th Monday
• Last Name AL F85
• Last Name MZ G60
• Todays Material
• Expected value and variance
• Bivariate Distributions

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Laws of Expected Value Variance

• Example 7.4
• The monthly sales at a computer store have a mean
  of 25,000 and a standard deviation of 4,000.
• Profits are 30 of the sales less fixed costs of
  6,000.
• Find the mean and standard deviation of the
  monthly profit.

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Example

• An electronics store sells two types of TVs
• Let X the number of small TV sets sold per day.
• Let Ythe number of large TV sets sold per day.
• Profit for a small TV set 200.
• Profit for a large TV set 400.
• Daily operating cost 250.
• Total Profit 200X 400Y 250.
• Assume X and Y are independent

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• Find the mean and standard deviation of total
  profits.
• How many more small TVs than large TVs should the
  store expect to sell in a day?
• c) What is the probability that the store will
  sell more small TVs than large TVs in a day?

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Example (cont)

• Now assume we are given the joint probability
  distribution of X and Y.

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Questions

• a) Calculate and interpret the coefficient of
  correlation between X and Y.
• b) Find the mean and SD of total TVs sold.
• c) Find the mean and SD of total profits.
• d) How did your answers to (b) and (c) change
  from before?

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Example

• Consider the following joint distribution.
• a) Find the probability distribution of
• b) Find the conditional distribution of X given
  that z-1.

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Example (cont)
a)
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Example (cont)
b)
So the conditional probability distribution of
X given that Z-1 is
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Example

• A quarter is flipped two times. Let X denote the
  number of heads on the first flip and Y denote
  the total number of heads on both flips. Find
  the correlation between X and Y.

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• Expected value and variance of X and Y.

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Example (Hard Problem??)

• A woman is expecting her second child. Her doctor
  has told her that she has a 50-50 chance of
  having another girl. If she has another girl,
  there is a 90 chance that she will be taller
  than the first. If she has a boy, however, there
  is only a 25 chance that he will be taller than
  the first child. Find the probability that the
  womans second child will be taller than the
  first.

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