{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

02_08ans - STAT 410 Examples for Spring 2008 Covariance of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
STAT 410 Examples for 02/08/2008 Spring 2008 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 ) . 1. Find in terms of σ X 2 , σ Y 2 , and σ XY : a) Cov ( 2 X + 3 Y , X – 2 Y ) , Cov ( 2 X + 3 Y , X – 2 Y ) = 2 Var ( X ) – Cov ( X , Y ) – 6 Var ( Y ) . b) Var ( 2 X + 3 Y ) , Var ( 2 X + 3 Y ) = Cov ( 2 X + 3 Y , 2 X + 3 Y ) = 4 Var ( X ) + 12 Cov ( X , Y ) + 9 Var ( Y ) . c) Var ( X – 2 Y ) . Var ( X – 2 Y ) = Cov ( X – 2 Y , X – 2 Y ) = Var ( X ) – 4 Cov ( X , Y ) + 4 Var ( Y ) .
Background image of page 1

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

View Full Document Right Arrow Icon
2. Consider the following joint probability distribution p ( x , y ) of two random variables X and Y: y x 0 1 2 p X ( x ) Recall: 1 0.15 0.15 0 0.30 E ( X ) = 1.7 2 0.15 0.35 0.20 0.70 E ( Y ) = 0.9 p Y ( y ) 0.30 0.50 0.20 E ( X Y ) = 1.65 Find Cov ( X , Y ) = σ XY . Cov ( X , Y ) = 1.65 – 1.7 0.9 = 0.12 . 3. Let the joint probability density function for ( X , Y ) be ( ) ° ± ² + = otherwise 0 1 , 1 0 , 1 0 24 , y x y x y x y x f Recall: E ( X ) = 0.40, E ( Y ) = 0.40. Find Cov ( X , Y ) = σ XY . E ( X Y ) = ³ ³ ´ ´ µ · · ¸ ¹ - 1 0 1 0 24 dx dy y x y x x = ( ) ³ - 1 0 3 2 1 8 dx x x = ³ ´ µ · ¸ ¹ - + - 1 0 5 4 3 2 8 24 24 8 dx x x x x = 6 8 5 24 6 3 8 - + - =
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}