hw5 - X and Y be given by f X,Y x,y = ± e-x ≤ y ≤ x<...

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

Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 5 Problem 1. The random variables X and Y have joint density f X,Y ( x,y ) = ± e - x 0 x < , 0 y 1 0 otherwise Evaluate the probability P [ X Y ]. Problem 2. X and Y are independent and uniform in the interval (0 ,a ). Find the pdf of Z = X/Y . Problem 3. Let the random variables X and Y have joint density f X,Y ( x,y ) = ( x y ln 2 exp ² - x 2 2 ³ 0 x < , 1 y 2 0 otherwise Find the probability density of Z = X/Y . Problem 4. X and Y are independent uniform random variables in the common interval (0 , 1). Determine f Z ( z ), where Z = X + Y . Problem 5. X and Y are independent, random variables with common pdf f X ( x ) = e - x U ( x ) f Y ( y ) = e - y U ( y ) Find the pdf of the following random variables (a) X - Y , (b) XY , and (c) min( X,Y ). Problem 6. The joint pdf of the random variables X and Y is given by f XY ( x,y ) = ± 1 shaded area Fig. 1 0 otherwise Let Z = X + Y . Find F Z ( z ) and f Z ( z ). Problem 7. Let the joint pdf of

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: X and Y be given by f X,Y ( x,y ) = ± e-x ≤ y ≤ x < ∞ otherwise 1 Figure 1: Problem 6. Deﬁne Z = X + Y , W = X-Y . Find the joint pdf of Z and W . Show that Z is not an exponential random variable. Problem 8. Let f X,Y ( x,y ) = ± 2 e-( x + y ) < x < y < ∞ otherwise Deﬁne Z = X + Y , W = Y/X . Determine the joint pdf of Z and W . Problem 9. (a) Use the auxiliary variable method to ﬁnd the pdf of Z = X X + Y . (b) Find the pdf of Z if X and Y are independent exponential random variables with the same parameter α . Problem 10. Suppose X and Y are zero-mean independent Gaussian random variables with common variance σ 2 . Deﬁne R = √ X 2 + Y 2 and θ = tan-1 ( Y/X ), where | θ | < π . Obtain their joint density function. 2...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

hw5 - X and Y be given by f X,Y x,y = ± e-x ≤ y ≤ x<...

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

View Full Document
Ask a homework question - tutors are online