HW13 - University of Illinois Fall 2009 ECE 313: Problem...

Info iconThis preview shows pages 1–2. 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: University of Illinois Fall 2009 ECE 313: Problem Set 13: Solutions Functions of Random Variables 1. [A piece of cake? or a sheet cake with a piece missing?] (a) The pdf is nonzero over the shaded region in the left-hand figure shown below. 1 / u v 1 1 1 / 2 1 / 2 u v 1 1 1 / 2 1 / 2 u v 1 1 1 / 2 1 / 2 (b) f Y ( v ) = Z - f X , Y ( u,v ) du = Z 1 1 / 2 4 3 du = 2 3 , < v 1 2 , Z 1 4 3 du = 4 3 , 1 2 < v < 1 , and f Y ( v ) = 0 otherwise. Use the horizontal lines in the middle figure above if you have difficulty figuring out where the limits came from. By symmetry, X has the same marginal pdf as Y . (c) Y X if the random point lies in the deep-shaded region in the lower right corner of the rightmost figure above. The area is 2- 8 = 3 8 and hence P { Y X } = 4 3 3 8 = 2 . (d) Y > X if the random point lies in the deep-shaded region in the upper left corner of the rightmost figure above. The area is 1 2 - 1 8 = 3 8 and hence P { Y X } = 1- 4 3 3 8 = 1- 1 2 . (e) P { Z } = P { Y X } = 2 , < 1 , 1- 1 2 , 1 < < . (f) f Z ( ) = d d F Z ( ) = d d P { Z } = 1 2 , < 1 , 1 2 2 , 1 < < , , elsewhere. Note that Z has the same pdf as I + 1 (with I = 1) that you found in Problem 3 of Problem Set 11, that is, with I = 1, f Z ( ) = f I ( - 1)....
View Full Document

Page1 / 3

HW13 - University of Illinois Fall 2009 ECE 313: Problem...

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