Chapter 5 Joint Probability Distribution Cont'

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Chapter 5Joint Probability Distribution
(Cont.)
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Covariance and correlation

• When two or more random variables are defined on
  a probability space, it is useful to describe how
  they vary together, i.e., measure the
  relationship between the variables.
• A common measure of the relationship between two
  random variables is the covariance.

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Covariance and correlation (Cont.)

• The expected value of a function of two random
  variables h(X, Y ), for discrete and continuous
  variable respectively
• Eh(X, Y) can be thought of as the weighted
  average of h(x, y) for each point in the range of
  (X,Y). It represents the average value of h(X,Y)
  that is expected in a long sequence of repeated
  trials of the random experiment.

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Example 5-27 Joint probability distribution for
X and Y
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Covariance

• The covariance between the random variables X and
  Y, denoted as cov(X,Y) or ?XY, is
• ?XY E(X - ?X)(Y - ?Y) E(XY) - ?X ?Y
• Covariance is a measure of the linear
  relationship between random variables.

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Covariance (Cont.)

• If the points in the joint probability
  distribution of X and Y that receive positive
  probability tend to fall along a line of positive
  (or negative) slope, ?XY is positive (or
  negative).
• If the relationship between the random variables
  is nonlinear, the covariance might not be
  sensitive to the relationship.

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Correlation

• The correlation between random variables X and Y,
  denoted as ?XY, is
• The correlation scales the covariance by the
  standard deviation of each variable.
• It is dimensionless quantity that can be sued to
  compare the linear relationship between pairs of
  variables in different units.

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Correlation (Cont.)

• Because ?X gt 0 and ?Y gt 0, if the covariance
  between X and Y is positive, negative, or zero,
  the correlation between X and Y is positive,
  negative, or zero respectively.
• If the points in the joint probability
  distribution of X and Y that receive positive
  probability tend to fall along a line of positive
  (or negative) slope, ?XY is near 1 (or -1).

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Correlation (Cont.)

• For any two random variables X and Y
• -1 ? ?XY ? 1
• If ?XY equals 1 or -1, the points in the joint
  probability distribution that receive positive
  probability fall exactly along a straight line.
• Two random variables with nonzero correlation are
  said to be correlated.

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Example 5-29 An example for a correlation that
is close to unity
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Example 5-30 An example of unity correlation
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Independence

• If X and Y are independent random variables,
• ?XY ?XY 0

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Linear combinations of random variables

• If the random variables X1 and X2 denote the
  length and width, respectively, of a manufactured
  part, Y 2X1 2X2 is a random variable that
  represents the perimeter of the part.
• Given random variables X1, X2, .., Xp and
  constants c1, c2, .., cp,
• Y c1X1 c2X2 cpXp
• is a linear combination of X1, X2, .., Xp.

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Linear combinations of random variables (Cont.)

• Example 5-37
• Reproductive property of the Normal distribution
• If X1, X2, , Xp are independent, normal random
  variables with E(Xi) ?i and V(X) ?2 i , for
  i 1, 2, .., p,
• Y c1X1 c2X2 cpXp
• is a normal random variable with
• E(Y) c1 ?1 c2 ?2 cp ?p
• and
• V(Y) c12 ?21 c22 ?2 2 cp2 ?2 p

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Mean and variance of an average

• If (X1 X2 Xp) / p with E(Xi) ?
  for i 1, 2, , p
• ?x
• If X1, X2, .., Xp are also independent with V(Xi)
  ?2 for i 1, 2, , p,
• ?2x / p
• Example 5-38

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ANNOUNCEMENTS

• Assignment IV
• 5, 12, 38, 52, 78, 89
• Due on