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