ECE 534 Fall 2009
August 27, 2009
Homework Assignment 1
Due Date: Thursday, September 10 (in class)
Announcement: There will be a probability review quiz on Tuesday, September 15 from 8:00-9:30
PM (the room will be announced later). The quiz will be closed book and you will not be allowed any
notes. Also, calculators, laptop computers, PDA’s, etc. are not permitted.
Reading: Chapter 1 of text (and if necessary, the notes on the ECE 313 website). Also read the
solutions to the even numbered problems of Chapter 1 given at the end of the book.
1. Tung, our beloved TA, has two coins in his pocket: one is a fair coin with a head and a tail, the
other is a special coin with two heads. Tung picks out a coin randomly, tosses it and observes
if he gets a head or a tail. After that, he puts the coin back into the pocket and repeats the
procedure once more. Specify the sample space Ω, the event space F, and the probability measure
Passigned to each event in F. What is the probability that a head was obtained in the first toss
given that the second toss is a tail?
2. Problem 1.3 of text.
3. Consider a communication system in which information is transmitted in bytes (sequences of 8 bits)
and all bytes are equally probable. A byte bis represented by the sequence of bits [b1, b2, . . . , b8],
where bi∈ {0,1}. The weight of a byte is the number of 1’s in the byte, i.e.,
w(b) =
8
X
i=1
bi
Define the following events:
A={b:b1=b2= 1},and B={b:w(b) is odd}
Evaluate P(A), P(B), P(B|A), and P(A|B).
4. Problem 1.7 of text.
5. Let U,V,Wbe independent, exponentially distributed random variables with parmeter α. Hence
each variable has mean 1
αand variance 1
α2.
(a) Evaluate E[W2].
(b) Evaluate E[UV W + 3U2].
(c) Evalulate Cov(2 + U, U V ).
6. Problem 1.11 of text.
7. Problem 1.21 of text.
c
V. V. Veeravalli, 2009 1
