UCSD ECE 153 Handout #18
Prof. Young-Han Kim Thursday, November 6, 2008
Homework Set #5
Due: Thursday, November 20, 2008
1. Work on the midterm problems to make sure you understand everything clearly.
2. Read Sections 6.1ā6.5 in the text. Try to work on all examples.
3. Which of the following matrices can be a covariance matrix? Justify your answer either
by constructing a random vector X, as a function of the i.i.d. zero mean unit variance
random variables Z1, Z2,and Z3, with the given covariance matrix, or by establishing
a contradiction.
(a) Ā·1 2
0 2 Āø(b) Ā·2 1
1 2 Āø(c) ļ£®
ļ£°
111
122
123
ļ£¹
ļ£»(d) ļ£®
ļ£°
112
123
233
ļ£¹
ļ£»
4. Given a Gaussian random vector Xā¼N(Āµ,Ī£), where Āµ= (1 2 3)Tand
Ī£ = ļ£®
ļ£°
900
041
011
ļ£¹
ļ£».
(a) What is P(X1+X2+ 2X3<0)?
(b) Find the joint pdf on Y=AX, where
A=Ā·1 1 ā1
2ā1 1 Āø.
5. Packet switching. Let Nā¼P(Ī»), i.e., Poisson with parameter Ī», be the number of
packets arriving at a switch per unit time. Each packet is routed to Output Port 1
with probability pand to Output Port 2 with probability (1 āp) independent of N
and of other packets. Let Xbe the number of packets routed to Output Port 1 per
unit time. Thus X= 0 if N= 0 and X=PN
i=1 Zifor N > 0, where
Zi=Ā½1,packet irouted to Port 1
0,packet irouted to Port 2,
and Z1, Z2, . . . , ZNare conditionally independent given N.
(a) Find the mean and variance of X.
(b) Find the pmf of X. What is the pmf of NāX?
1
