Example_class_9_handout0

# Example_class_9_handout0 - aX + c,bY + d ) = sign ( ab )...

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

THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 9 Review Covariance Let X and Y be random variables with mean μ x and μ y respectively. The covariance between X and Y, denoted by σ xy , is deﬁned as Cov ( X,Y ) = E [( X - μ x )( Y - μ y )] = E ( XY ) - E ( X ) E ( Y ) Properties: (a) Cov ( aX + c,bY + d ) = abCov ( X,Y ) In general, Cov ( n X i =1 a i X i + c i , N X j =1 b j Y j + d j ) = m X i =1 n X j =1 a i b j Cov ( X i ,Y j ) (b) Cov ( X,X ) = V ar ( X ) (c) V ar ( X + Y ) = V ar ( X ) + V ar ( Y ) + 2 Cov ( X,Y ) In general, V ar ( n i =1 X i ) = n i =1 V ar ( X i ) + 2 i<j Cov ( X i ,Y j ) (d) If X and Y are independent, then Cov ( X,Y ) = 0 But Cov ( X,Y ) = 0 does not imply X and Y are indepenent. (Very important!) Correlation Coeﬃcient Let X and Y be two random variables. The correlation coeﬃcient between X and Y, denoted as ρ xy is deﬁned as Corr ( X,Y ) = Cov ( X,Y ) p V ar ( X ) p V ar ( Y ) Properties: (a) - 1 ρ 1 (b) ρ is invariant under linear transformation X and Y. That is, Corr (

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.

Unformatted text preview: aX + c,bY + d ) = sign ( ab ) Corr ( X,Y ) The sign and the magnitude of ρ reveal the direction and strength of the linear relationship between X and Y. Sum of independent random variables Let X 1 ,X 2 ,...,X n be independent random variables, Y = ∑ n i =1 a i X i , where a i s are constants. Then, E ( Y ) = ∑ n i =1 aE ( X i ) V ar ( Y ) = ∑ n i =1 V ar ( a i X i ) = ∑ n i =1 a 2 i V ar ( X i ) M y ( t ) = Q n i =1 M x i ( a i t ) 1 Problems Problem 1 Let X and Y be two discrete random variables with joint probability mass function shown in the table. (a)Calculate the covariance and correlation coecient between X and Y. (b)Determine whether X and Y are independent. 2 3...
View Full Document

## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

### Page1 / 3

Example_class_9_handout0 - aX + c,bY + d ) = sign ( ab )...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online