Chapter 5, Sections 78 Multivariate Distributions

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Chapter 5, Sections 7-8 Multivariate
Distributions
Covariance Expected Value of Functions of RVs
? John J Currano, 03/30/2009
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Definitions. Let Y1 and Y2 be random variables
with means ?1 and ?2, respectively. We define the
covariance of Y1 and Y2 to be Cov(Y1, Y2) E
(Y1 ? ?1) (Y2 ? ?2) and we define the
correlation cofficient, ?? , of Y1 and Y2 to be
Theorem 5.10. (p. 266) Cov(Y1, Y2) E (Y1Y2) ?
E (Y1) E (Y2). This theorem is proved in the
text. The proof only uses the properties of
expectation from Section 5.6 (Theorems 5.6 5.8).
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Theorem 5.10. Cov(Y1, Y2) E (Y1Y2) ? E (Y1) E
(Y2).
Example. Suppose
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Theorem 5.10. Cov(Y1, Y2) E (Y1Y2) ? E (Y1) E
(Y2).
Example. Suppose
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f (y1, y2) 6(1 y2) for 0 ? y1 ? y2 ? 1
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Since Cov(Y1, Y2) E (Y1Y2) ? E (Y1) E (Y2),
and since if Y1 and Y2 are independent E (Y1Y2)
E (Y1) E (Y2), we have the following
theorem Theorem. If Y1 and Y2 are independent
random variables, then Cov(Y1, Y2)
0 and Corr(Y1, Y2) ? 0. The converse is
false, as the next example shows.
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Example. Suppose X U (??1, 1) and Y X 2.
What are Cov (X, Y ) and Corr (X, Y )? Using the
computing formula for covariance, Cov(X, Y) E
(X Y) ? E (X) E (Y). Since X U (??1, 1),
Since Y X 2, Since X Y X 3, so Cov (X, Y )
E (X Y) ? E (X) E (Y) 0 0 ? (1/3)
0 and Thus X and Y are uncorrelated even though
Y X 2 !
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Properties of the Covariance Operator Recall
Cov (Y1, Y2) E (Y1 ? ?1) (Y2 ? ?2) .
Theorem 1. Cov (Y1, Y2) Cov (Y2, Y1). Theorem
2. V (Y ) Cov (Y, Y). Theorem 3. Cov(Y1, Y2)
E (Y1Y2) ? E (Y1) E (Y2). Theorem 4. Cov
(a1Y1, a2Y2) a1a2 Cov (Y1, Y2). Theorem 5. Cov
(X, Y1 Y2 ) Cov (X, Y1 ) Cov (X, Y2 ).
Cov (Y1 Y2, X ) Cov (Y1, X ) Cov (Y2, X
). Theorem 6. (Bilinearity of the Covariance
Operator)
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Theorem 6. (Bilinearity of the Covariance
Operator) Theorem 7. (Variance of a linear
combination of random variables) Theorem 8.
(Variance of a linear comb. of independent random
vars.) If Y1, Y2, . . . , Yn are independent
random variables, then
Note Theorems 6 7 are parts of Theorem 5.12 on
page 271 of the text.
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The correlation cofficient provides a measure of
both the sign and the degree of the linear
association between Y1 and Y2 . Since
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• ? gt 0 means Y1 and Y2 tend to be above average
  together and below average together
• (Y1 ? ?1) and (Y2 ? ?2) tend to have same
  sign.
• ? lt 0 indicates that one of Y1 and Y2 tends to
  be above average when the other is
  below average
• (Y1 ? ?1) and (Y2 ? ?2) tend to have
  opposite signs.
• the absolute value of ? (between 0 and 1) is a
  measure of the strength of the association.

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• The correlation cofficient
• has the following properties
• ?1 ? ? ? 1
• If Y2 aY1 b (so that Y1 and Y2 are linearly
  related), then
• If ? 1, then Y1 and Y2 are linearly
  related with a gt 0 if ? 1 and a lt 0 if ? ?1.
• ? 0 ?? Cov(Y1, Y2) 0 ? E (Y1 Y2) E (Y1)
  E (Y2).
• If ? 0, we say that Y1 and Y2 are
  uncorrelated. They may or may not be independent.

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Uncorrelated random variables (? 0) need not be
independent, although independent random
variables are uncorrelated since Theorem 5.9 ?
E (Y1 Y2) E (Y1) E (Y2) when Y1 and Y2 are
independent, and then property 4 on the previous
slide ? ?? 0. The following is an example of
two uncorrelated random variables which are not
independent
Similar to Table 5.3 on page 267.
Then E (Y1) E (Y2) E (Y1 Y2) 0, but Y1
and Y2 are not independent, as can be seen since
the joint support is not a rectangle or p (?1,
0) 0, while
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Example. Consider We have already calculated
Find (a) E(2Y1 3Y2) (b) V(2Y1 3Y2)
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Example. Consider We know
Find (a) E(2Y1 3Y2)

• 2 E(Y1) 3 E(Y2)
• ?1

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Example. Consider We know
Find (b) V ( 2Y1 3Y2) V ( 2Y1 3Y2) 22
V(Y1) ( 3)2 V(Y2) 2(2)(3) Cov(Y1, Y2)
4 V(Y1) 9 V(Y2) 12
Cov(Y1, Y2) We need to find V(Y1) and V(Y2).
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Example. Consider We know
We need to find V(Y1) and V(Y2)
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Example. Consider We have found
Thus, V(2Y1 3Y2) ?? 4 V(Y1) ?? 9 V(Y2)
? 12 Cov(Y1, Y2)
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Read Examples 5.27 5.29 (pages 274-276). These
present three important results.
Example 5.27. If Y1, Y2, . . . , Yn are
independent random variables with common mean ??
and variance ? 2, and if
Where this is used Y1, Y2, . . . , Yn are the
outcomes of n independent trials of an
experiment, called a Random Sample. is
called the Sample Mean.
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• Example 5.28. If
• p P(success) in a sequence of independent
  binomial trials
• Y binomial(n, p)
• an estimate of p
• then

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Example 5.29. Y, the number of red balls in a
sample of size n selected at random without
replacement from an urn with r red and (N r)
black balls, is a hypergeometric random variable.
On their way to calculating the mean and
variance of Y, they show that the probability a
red is chosen on any given draw is This is
obvious for the first draw, but not for the rest.