zz-23 - 7. cov ( aX + b, cY + d ) = accov ( X, Y ) cov ( X,...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 7. cov ( aX + b, cY + d ) = accov ( X, Y ) cov ( X, Y ) = E [( X- E [ X ])( Y- E [ Y ])] = E [(( aX- b )- E ( aX + b ))(( cY + d )- E ( cY + d ))] E ( aX + bY ) = aE [ X ] + bE [ Y ] cov ( aX + b, cY + d ) = ( aX- b- aE [ X ] + b )( cY + d- cE [ Y ] + d ) E [( aX- aμ x )( cY- cμ y )] = E [ a ( x- μ x ) c ( Y- μ y )] cov ( aX + b, cY + d ) = ac E [( X- μ x )( Y- μ y )] | {z } cov ( X,Y ) = accov ( X, Y ) Q.E.D 8. X ∼ N (0 , sigma 2 ) Φ X ( μ ) = E [ e iμx ] = Z ∞-∞ e iμx f x ( x ) dx f x ( x ) = 1 √ 2 π 2 σ x e- ( x- μx ) 2 2 σ 2 x Φ x ( μ ) = e iμu- 1 2 u 2 σ 2 = e- 1 2 u 2 σ 2 E [ X k ] = i- k d k du k Φ x ( u ) | u =0 = i- k d k du k e- 1 2 u 2 σ 2 | u =0 = d k du k i- k (1- x + x 2 2!- x 3 3! ... ) | u =0 = d k du k i- k (1- σ 2 u 2 2 1 2! + σ 4 u 4 2 2 2!- σ 6 u 6 2 3 3! + σ 8 u 8 2 4 4! ... ) | u =0 E [ x k ] = when k is odd σ k k ! 2 k 2 ( k 2 )! when k is even 1 1.4.4 Solution: The final calculation of 2 3 refers not to a single draw of one ball from an urn containing three, but rather to a composite experiment comprising more than one stage. While it is true that { two black, one white } is the only fixed collection of balls for which a random choice is black with probability 2 3 , the composition of the urn is not determined...
View Full Document

This note was uploaded on 03/01/2012 for the course ECE 6010 taught by Professor Stites,m during the Spring '08 term at Utah State University.

Page1 / 6

zz-23 - 7. cov ( aX + b, cY + d ) = accov ( X, Y ) cov ( X,...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online