{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 09_07 - STAT 410 Examples for Fall 2011 2.4 Covariance and...

This preview shows pages 1–4. Sign up to view the full content.

STAT 410 Examples for 09/07/2011 Fall 2011 2.4 Covariance and Correlation Coefficient Covariance of X and Y σ XY = Cov ( X , Y ) = E [ ( X – μ X ) ( Y – μ Y ) ] = E ( X Y ) μ X μ Y (a) Cov ( X , X ) = Var ( X ) ; (b) Cov ( X , Y ) = Cov ( Y , X ) ; (c) Cov ( a X + b , Y ) = a Cov ( X , Y ) ; (d) Cov ( X + Y , W ) = Cov ( X , W ) + Cov ( Y , W ) . Cov ( a X + b Y , c X + d Y ) = a c Var ( X ) + ( a d + b c ) Cov ( X , Y ) + b d Var ( Y ) . Var ( a X + b Y ) = Cov ( a X + b Y , a X + b Y ) = a 2 Var ( X ) + 2 a b Cov ( X , Y ) + b 2 Var ( Y ) . 0. Find in terms of σ X 2 , σ Y 2 , and σ XY : a) Cov ( 2 X + 3 Y , X – 2 Y ) , b) Var ( 2 X + 3 Y ) , c) Var ( X – 2 Y ) .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Correlation coefficient of X and Y ρ XY = Y X XY σ σ σ = ( ) ( ) ( ) , Y Var X Var Y X Cov = - - Y Y , X X σ μ σ μ Y X E (a) 1 ρ XY 1; (b) ρ XY is either + 1 or – 1 if and only if X and Y are linear functions of one another. If random variables X and Y are independent, then E ( g ( X ) h ( Y ) ) = E ( g ( X ) ) E ( h ( Y ) ) . Cov ( X, Y ) = σ XY = 0 , Corr ( X , Y ) = ρ XY = 0 . 1. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y Recall: E ( X ) = 1.75, E ( Y ) = 0.8, E ( X Y ) = 1.5. x 0 1 2 p X ( x ) 1 0.15 0.10 0 0.25 2 0.25 0.30 0.20 0.75 p Y ( y ) 0.40 0.40 0.20 1.00 Find Cov ( X , Y ) = σ XY and Corr ( X , Y ) = ρ XY .
2. Let the joint probability density function for

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

09_07 - STAT 410 Examples for Fall 2011 2.4 Covariance and...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online