Title: Lecture 15: Expectation for Multivariate Distributions
1Lecture 15 Expectation for Multivariate
Distributions
Those who ignore Statistics are condemned to
reinvent it. Brad Efron
- Probability Theory and Applications
- Fall 2008
2Outline
- Correlation
- Expectations of Functions of R.V.
- Covariance
- Covariance and Independence
- Algebra of Covariance
3Correlation 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/
4Sometimes not so perfect
- Arm Strength Versus Grip Strength
Pearsons Correlation R.63
5Negative Correlation
6Extreme Correlation 1
- Linear relation with positive slope
7Extreme Correlation -1
- Linear relation with negative slope
8Zero Correlation
9Guess 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
10Expectations of Functions of R.V.
NOTE substitute appropriate summation for discrete
11Variance and Covariance
- Univariate becomes variance
- Multivariate becomes covariance
- Note
NOTE substitute appropriate integral for
continuous
12Calculating Covariance
13Correlation of X and Y
- Definition
- The correlation always falls in -1, 1
- It a measure of the linear relation between X and
Y
14Extreme Cases
- If XY then ?1.
- If X-Y then ?-1.
- If X and Y independent, then ?0.
- If X-2Y then ??.
15Example
- Joint is
- Find correlation of X and Y
16Example
- Joint is
- Find correlation of X and Y
17Properties of Covariance
- a)
- b) cov(aXbY)abcov(X,Y)
18Properties of Covariance
19Properties of Covariance
- If X and Y are independent
- Proof
20Find Covariance
X\Y 0 1
-1 0 .3
0 .4 0
1 0 .3
21Are X and Y independent
X\Y 0 1
-1 0 .3
0 .4 0
1 0 .3
22Note
- 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.
23Algebra of variance/covariance/correlation
- Given
- Calculate mean of Z2X-3Y5
- variance of Z2X-3Y5
24Long steps
25Working Rules for linear combinations
- Write formula
- Discard Constants
- Square it
- Replace squared R.V
- with var and crossterms
- with cov
26Example
- 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
28Working rule works also for more than two
variables
Find variance of W2x-3Y5Z1