Lecture 3: Combinatorics

Lecture 3: Combinatorics

Those who ignore Statistics are condemned to reinvent Brad Efron Lecture 15: Expectation for Multivariate Distributions Probability Theory and Applications Fall 2008 Outline

Correlation Expectations of Functions of R.V. Covariance Covariance and Independence Algebra of Covariance 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/ Sometimes not so perfect Arm Strength Versus Grip Strength Pearsons Correlation R=.63 Negative Correlation Child Labor versus GDP Extreme Correlation 1 Linear relation with positive slope

Extreme: Correlation -1 Linear relation with negative slope Zero Correlation Independent Random 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 Expectations of Functions of R.V. E ( g ( x, y )) g ( x, y ) f ( x, y )dydx

E ( X Y ) ( x y ) f ( x, y )dxdy x f ( x, y )dy dx y f ( x, y )dx dy

xf x ( x)dx yf y ( y )dy E ( X ) E (Y )

NOTE substitute appropriate summation for discrete Variance and Covariance Univariate becomes variance var( X ) E ( X X ) 2 ( x X ) 2 f X ( x) x Multivariate becomes covariance cov( X , Y ) E ( X X )( y Y ) ( x X )( y Y ) f ( x, y ) x, y

Note: var( X ) cov( X , X ) NOTE substitute appropriate integral for continuous Calculating Covariance Can simplify cov( X , Y ) E ( X X )(Y Y ) xy y X xY X Y f ( x, y ) x, y xy f ( x, y ) y X x Y f ( x, y ) X Y x, y

x, y E ( XY ) X Y X Y X Y E ( XY ) E ( X ) E (Y ) Correlation of X and Y Definition cov( X , Y ) ( X ,Y ) XY The correlation always falls in [ -1, 1] It a measure of the linear relation between X

and Y Extreme Cases If X=Y then =1. If X=-Y then =-1. If X and Y independent, then =0. If X=-2Y then =?. Example 2 x 0, y 0, x y 1 f ( x, y ) o.w. 0

Joint is Find correlation of X and Y 1 x 1 y fY ( y ) 2dx 2(1 y ) 0 y 1 f X ( x) 2dy 2(1 x) 0 x 1 0 0

1 1 E (Y ) 2 y (1 y ) dy 1/ 3 E ( X ) 2 x(1 x) dx 1/ 3 0 0 1 1

E (Y ) 2 y (1 y ) dy 1/ 6 E ( X ) 2 x 2 (1 x) dx 1/ 6 var(Y ) 1/ 6 1/ 9 1/18 var( X ) 1/ 6 1/ 9 1/18 2 2 0

2 0 Example Joint is 2 x 0, y 0, x y 1 f ( x, y ) o.w. 0 Find correlation of X and Y

1 1 y 1 E ( X , Y ) 2 xydxdy y (1 y ) 2 dy 1/12 0 0 0 cov( X , Y ) 1/12 1/ 9 1/ 36 cov( X , Y ) 1/ 36 ( X ,Y )

1/ 2 XY 1/18 1/18 Properties of Covariance a) var( X ) cov( X , X ) b) cov(aX+bY)=ab[cov(X,Y)] E (aXbY ) abE ( XY ) E (aX ) aE ( X ) E (bY ) bE (Y ) cov(aX bY ) E (aXbY ) E (aX ) E (bY ) ab[ E ( X , Y ) E ( X ) E (Y )] ab cov( X , Y ) Properties of Covariance

c) var(aX ) a 2 var( X ) d) (aX bY ) sign(a * b) ( X , Y ) ab cov( X , Y ) (aX bY ) ( X ,Y ) | a | X | b | Y ab ( X , Y ) sign(a * b) ( X , Y ) | a || b | Properties of Covariance e) If X and Y are independent cov( X , Y ) 0

correl ( X , Y ) 0 Proof: E ( X , Y ) xyf x ( x) f y ( y )dxdy xf x ( x)dx yfY ( y )dy E ( X ) E (Y ) cov( X , Y ) E ( XY ) E ( X ) E (Y ) 0 Find Covariance X\Y 0 1

-1 0 .3 0 .4 0 1

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

0 .3 0 .4 0 1 0

.3 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 Y=X2 so the variables are definitely not independent but their covariance is 0 because they have no linear relation. Algebra of variance/covariance/correlation

Given: E ( X ) Y var( X ) X2 E (Y ) Y var(Y ) Y2 cov( X , Y ) XY Calculate mean of Z=2X-3Y+5 variance of Z=2X-3Y+5 Long steps E ( X ) 2 X 3Y 5

Z E ( Z ) 2( X X ) 3(Y Y ) [ Z E ( Z )]2 4( X X ) 2 3(Y Y ) 2 12( X X )(Y Y ) E [ Z E ( Z )]2 4 var( X ) 8 var(Y ) 12cov( X , Y ) Working Rules for linear combinations Z 2 X 3Y 5 Write formula 2 X 3Y Discard Constants 2 2 4 X

9 Y 12 XY Square it Replace squared R.V 4 var( X ) 9 var(Y ) 12cov( X , Y ) with var and crossterms with cov Example Given var(X)=4 var(Y)=10 (X,Y)=1/2 Find variance of X-5Y+6?

Given same facts as previous problem Find covariance x-5y+6 and -4X+3Y+2 Working rule works also for more than two variables Find variance of W=2x-3Y+5Z+1

Recently Viewed Presentations

  • CXR's - prestonmedsoc.co.uk

    CXR's - prestonmedsoc.co.uk

    A 60 year old, Andrea Collins presents to her GP with fatigue, weight loss and wheeze. There is no significant past medical history. She is a life long smoker with saturations of 99% and is afebrile. There is a wheeze...
  • Lssm - Gahsc

    Lssm - Gahsc

    Data Driven system…uses the DFCS child welfare SACWIS system to monitor outcomes and as case record. Web posted dash boards. Competitive procurement process. Increased performance and state out comes on CFSR audit (shorter time out of home, increased adoptions, more...
  • Simple Machines: Gear Ratios and Velocity Ratios

    Simple Machines: Gear Ratios and Velocity Ratios

    Simple Machines: Gears, Velocity Ratios and Mechanical Advantage What is a simple machine? A device which uses basic mechanisms to make work easier for the user.
  • INTEGRATING EXPERIMENTAL DESIGN INTO SCIENCE Experimental Design: The

    INTEGRATING EXPERIMENTAL DESIGN INTO SCIENCE Experimental Design: The

    INTEGRATING EXPERIMENTAL DESIGN INTO SCIENCE Experimental Design: The Process Paper airplane - everybody builds one Observe the plane's flight Ready, set, hold it … How do we determine which is best? 5 minutes to modify, make one change Write your...
  • After Perfection

    After Perfection

    Cento concerti . ecclesiastici, a . una, a due, a . tre & a . quattro. voci (1602) Basso continuo. ... Setting of a Petrarch poem. Paradox and Contradiction:Late Italian Madrigalists. Claudio Monteverdi (1567-1643) Fifth book of Madrigals . Cruda....
  • HNG DN QUYT TON THU THU NHP DOANH

    HNG DN QUYT TON THU THU NHP DOANH

    HƯỚNG DẪN QUYẾT TOÁN THUẾ THU NHẬP DOANH NGHIỆP NĂM 2017 CHI CỤC THUẾ TP.THANH HÓA Tháng 03/2018 CĂN CỨ PHÁP LÝ 1 - Thông tư số 78/2014/TT-BTC ngày 18/06/2014 của Bộ Tài chính hướng dẫn thực hiện Nghị định...
  • Zoology - Arthropod Unit

    Zoology - Arthropod Unit

    Arthropod Phylogeny. Subphylum Trilobita. Extremely common in the early Paleozoic era, trilobites are extinct today. Subphylum Chelicerata. Class Merostomata includes Horseshoe Crabs. Class Pycnogonida includes Sea Spiders. Class Arachnida includes spiders, scorpions, ticks and mites. Subphylum Crustacea. Lobsters, crabs, shrimp,...
  • Theories of Deviance Strain Theory Theoretical Perspective: Functionalism

    Theories of Deviance Strain Theory Theoretical Perspective: Functionalism

    The Ratio of Deviant people to Non-Deviant people observed during ones life. Occurrence of Deviant behavior by ones significant others during ones life. The age to which one was exposed to deviant behavior. Three Characteristics: Differential Association Theory. Observing More...