MIT6_041F10_assn06_sol

MIT6_041F10_assn06_sol - Massachusetts Institute of...

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: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 6: Solutions 1. Let us draw the region where f X,Y ( x, y ) is nonzero: y 2 1 y-x=z 0 1 2 x x =2 y = x The joint PDF has to integrate to 1. From x =1 y =0 ax dy dx = 3 7 a = 1, we get a = 7 3 . 2 3 9 1 7 x dx, if 0 ≤ y ≤ 1 , , if 0 ≤ y ≤ 1 , 14 (a) b f Y ( y ) = f X,Y ( x, y ) dy = 2 3 x dx, if 1 < y ≤ 2 , = 3 (4 − y 2 ) , if 1 < y ≤ 2 , y 7 14 , otherwise , otherwise . (c) f X | Y ( x | 3 ) = f X,Y ( x, 2 3 ) = 8 x, for 3 2 ≤ x ≤ 2 and 0 otherwise . 2 f Y ( 3 2 ) 7 Then, 2 1 3 1 8 4 E | Y = = x dx = . X 2 3 / 2 x 7 7 (d) We use the technique of first finding the CDF and differentiating it to get the PDF. F Z ( z ) = P ( Z ≤ z ) = P ( Y − X ≤ z ) , if z < − 2 , x =2 y = x + z 3 8 6 3 x dy dx = + z − 1 z , if − 2 ≤ z ≤ − 1 , 7 7 14 7 x = − z y =0 = x =2 y = x + z 3 x dy dx = 1 + 9 z, if − 1 < z ≤ , 14 7 x =1 y =0 1 , if 0 < z. 6 − 3 z 2 , if − 2 ≤ z ≤ − 1 , d 7 14 9 f Z ( z ) = F Z ( z ) = 14 , if − 1 < z ≤ , dz , otherwise . Page 1 of 4 Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) 2. The PDF of Z , f Z ( z ), can be readily computed using the convolution integral: ∞ f Z ( z ) = f X ( t ) f Y ( z − t ) dt....
View Full Document

This note was uploaded on 01/11/2012 for the course EE 6.431 taught by Professor Prof.dimitribertsekas during the Fall '10 term at MIT.

Page1 / 5

MIT6_041F10_assn06_sol - Massachusetts Institute of...

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