HW8 - Mathematics Department, UCLA T. Richthammer winter...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematics Department, UCLA T. Richthammer winter 09, sheet 8 Feb 20, 2009 Homework assignments: Math 170A Probability, Sec. 1 093. The following functions are CDFs of RVs. Determine the corresponding PDF: (a) F ( x ) = 1 x 2 +1 1 ( -∞ , 0) ( x ) + 1 [0 , ) ( x ) (b) F ( x ) = e x 2 1 ( -∞ , 0) ( x ) + (1 - e - x 2 )1 [0 , ) ( x ) Answer: We use f ( x ) = F ± X ( x ), so (a) f ( x ) = - 2 x ( x 2 +1) 2 1 ( -∞ , 0) ( x ) (b) f ( x ) = 1 2 e -| x | . 094. Let X be a RV with PDF f ( x ) = 1 18 x 2 1 [ - 3 , 3] ( x ). Calculate R ( Y ) and the PDF of Y , where (a) Y = X + 2 (b) Y = 3 X (c) Y = e X (d) Y = X 2 (e) Y = - X Answer: R ( X ) = [ - 3 , 3] and for c R ( X ) we have F X ( c ) = 1 18 R c - 3 x 2 = c 3 +27 54 . (a) R ( Y ) = [ - 1 , 5] and for c [ - 1 , 5] we have F Y ( c ) = P ( X + 2 c ) = P ( X c - 2) = F X ( c - 2), so f Y ( c ) = f X ( c - 2) = 1 18 ( c - 2) 2 1 [ - 1 , 5] ( c ) (b) R ( Y ) = [ - 9 , 9] and for c [ - 9 , 9] we have F Y ( c ) = P (3 X c ) = P ( X c/ 3) = F X ( c/ 3), so f Y ( c ) = f X ( c/ 3) / 3 = 1 486 c 2 1 [ - 9 , 9] ( c ) (c) R ( Y ) = [ e - 3 , e 3 ] and for c R ( Y ) we have F Y ( c ) = P ( e X c ) = P ( X ln c ) = F X (ln c ), so f Y ( c ) = f X (ln c ) /c = 1 18 (ln c )2 c 1 [ e - 3 ,e 3 ] ( c ) (d) R ( Y ) = [0 , 9] and for c R ( Y ) we have F Y ( c ) = P ( X 2 c ) = P ( | X | ≤ c ) = F X ( c ) - F X ( - c ), so f Y ( c ) = f X ( c ) 1 2 c + f X ( - c ) 1 2 c = c 18 1 [0 , 9] ( c ) (e) R ( Y ) = [ - 3 , 3] and for c R ( Y ) we have F Y ( c ) = P ( - X c ) = P ( X ≥ - c ) = 1 - F X ( - c ), so f Y ( c ) = f X ( - c ) = 1 18 x 2 1 [ - 3 , 3] ( c ) = f ( c ) 095. For a > 0, b R express the PDF of Y = aX + b in terms of the PDF f X of the RV X . Answer: F Y ( c ) = P ( aX + b c ) = P ( X c - b a ) = F X ( c - b a ), so f Y ( c ) = 1 a f X ( c - b a ). 096. Compute
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/07/2010 for the course MATH 170A 170A taught by Professor Richthammer during the Winter '10 term at UCLA.

Page1 / 4

HW8 - Mathematics Department, UCLA T. Richthammer winter...

This preview shows document pages 1 - 2. Sign up to view the full document.

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