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Lecture 15: Expectation for Multivariate Distributions

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Title: Lecture 15: Expectation for Multivariate Distributions


1
Lecture 15 Expectation for Multivariate
Distributions
Those who ignore Statistics are condemned to
reinvent it. Brad Efron
  • Probability Theory and Applications
  • Fall 2008

2
Outline
  • Correlation
  • Expectations of Functions of R.V.
  • Covariance
  • Covariance and Independence
  • Algebra of Covariance

3
Correlation Intuition
  • Covariance is a measure of how much RV vary
    together.
  • Wifes Age and Husbands Age

Correlation .97
Example from http//cnx.org/content/m10950/latest/
4
Sometimes not so perfect
  • Arm Strength Versus Grip Strength

Pearsons Correlation R.63
5
Negative Correlation
  • Child Labor versus GDP

6
Extreme Correlation 1
  • Linear relation with positive slope

7
Extreme Correlation -1
  • Linear relation with negative slope

8
Zero Correlation
  • Independent Random

9
Guess Covariance???Positive, Negative, 0
  • Crime Rate, Housing Price
  • SAT Scores, GPA Freshman Year
  • Weight and SAT Score
  • Average Daily Temperature, Housing Price
  • GDP, Infant Mortality
  • Life Expectancy, Infant Mortality

10
Expectations of Functions of R.V.
NOTE substitute appropriate summation for discrete
11
Variance and Covariance
  • Univariate becomes variance
  • Multivariate becomes covariance
  • Note

NOTE substitute appropriate integral for
continuous
12
Calculating Covariance
  • Can simplify

13
Correlation of X and Y
  • Definition
  • The correlation always falls in -1, 1
  • It a measure of the linear relation between X and
    Y

14
Extreme Cases
  • If XY then ?1.
  • If X-Y then ?-1.
  • If X and Y independent, then ?0.
  • If X-2Y then ??.

15
Example
  • Joint is
  • Find correlation of X and Y

16
Example
  • Joint is
  • Find correlation of X and Y

17
Properties of Covariance
  • a)
  • b) cov(aXbY)abcov(X,Y)

18
Properties of Covariance
  • c)
  • d)

19
Properties of Covariance
  • If X and Y are independent
  • Proof

20
Find Covariance
X\Y 0 1
-1 0 .3
0 .4 0
1 0 .3

21
Are X and Y independent
X\Y 0 1
-1 0 .3
0 .4 0
1 0 .3

22
Note
  • Cov(X,Y)0 does not imply independence of X and Y
  • Independence of X and Y implies cov(X,Y)0
  • In this case YX2 so the variables are
    definitely not independent but their covariance
    is 0 because they have no linear relation.

23
Algebra of variance/covariance/correlation
  • Given
  • Calculate mean of Z2X-3Y5
  • variance of Z2X-3Y5

24
Long steps
25
Working Rules for linear combinations
  • Write formula
  • Discard Constants
  • Square it
  • Replace squared R.V
  • with var and crossterms
  • with cov

26
Example
  • Given var(X)4 var(Y)10
  • ?(X,Y)1/2
  • Find variance of X-5Y6?

27
  • Given same facts as previous problem
  • Find covariance x-5y6 and -4X3Y2

28
Working rule works also for more than two
variables
Find variance of W2x-3Y5Z1
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