hw3sol - APPM 4/5520 Solutions to Problem Set Three 1. (a)...

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: APPM 4/5520 Solutions to Problem Set Three 1. (a) f ( x ) = p (1- p ) x I { , 1 , 2 ,... } ( x ) M X ( t ) = E [ e tX ] = x =0 e tx p (1- p ) x = p x =0 bracketleftbig e t (1- p ) bracketrightbig x This geometric sum converges if e t (1- p ) < 1. (Note that it is already positive since e t > 0 and since 0 p 1 for the geometric pdf.) So, we have convergence if e t < (1- p ) 1 which implies that t < ln(1- p ) 1 =- ln(1- p ). Using the fact that x =0 r n = 1 1 r for | r | < 1, we have that M X ( t ) = p summationdisplay x =0 bracketleftBig e t (1- p ) bracketrightBig x = p 1 1- e t (1- p ) = p 1- e t (1- p ) for t <- ln(1- p ). (b) Y = X + 1 So, M Y ( t ) = E [ e tY ] = E [ e t ( X + 1)] = e t E [ e tX ] = e t M X ( t ) = pe t 1- e t (1- p ) for t <- ln(1- p ). 2. The moment generating function for X ( , ) is M ( t ) = bracketleftbigg - t bracketrightbigg for t < . So, M ( t ) = bracketleftBig t bracketrightBig 1 d dt bracketleftBig t bracketrightBig = bracketleftBig t bracketrightBig 1 ( t ) 2 and M ( t ) = d dt braceleftbigg bracketleftBig t bracketrightBig 1 ( t ) 2 bracerightbigg = bracketleftBig t bracketrightBig 1 2 ( t ) 3 + ( t ) 2 ( - 1) bracketleftBig t bracketrightBig 2 ( t ) 2 . Now, E [ X ] = M (0) = bracketleftbigg - bracketrightbigg 1 ( - 0) 2 = , as desired, and E [ X 2 ] = M (0) = bracketleftBig bracketrightBig 1 2 ( 0) 3 + ( 0) 2 ( - 1) bracketleftBig bracketrightBig 2 ( 0) 2 = 2 + 2 ....
View Full Document

This note was uploaded on 02/07/2012 for the course APPM 4520 taught by Professor Manuel during the Fall '11 term at University of Colorado Denver.

Page1 / 6

hw3sol - APPM 4/5520 Solutions to Problem Set Three 1. (a)...

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