ECE/CS 541: COMPUTER SYSTEM ANALYSIS
Homework #1 – Due Tuesday, September 1, Beginning of Class
August 25, 2008
The intent of this homework is to review material you should have learned in your probability and statis-
tics course (e.g., ECE 413) and explore new material regarding probability theory and stochastic proc-
esses we have discussed in class. They vary in difficulty from those you should be able to solve in a cou-
ple of minutes, to those that require more thought. If you have trouble with these problems, you should
review material from your probability course, or come and talk to me during office hours.
1. Consider the following program segment:
if B then
repeat S1 until B1
else
repeat S2 until B2
Assume that P(B = true) = p, P(B1 = true) = 5/8, and P(B2 = true) = 3/8. Exactly one statement is
common to statement groups S1 and S2: write (‘good day’). After many repeated executions of the
program segment, it has been estimated that the probability of printing exactly three ‘good day’ mes-
sages is 15/128. Derive the value of p.
2. In manufacturing a certain component, two types of defects are likely to occur with respective prob-
abilities 0.05 and 0.1. Assuming that the defects are independent of one another, what is the probabil-
ity that a randomly chosen component:
(a) Does not have both defects?
(b) Is defective?
(c) Has only one kind of defect, given that it is found to be defective?
3. A mischievous student wants to break into a computer file, which is password-protected. Assume
that there are n equally likely passwords, and that the student chooses passwords independently and at
random and tries them. Let Nn be the number of trials required to break into the file. Determine the
probability density function of Nn (a) if unsuccessful passwords are not eliminated from further selec-
tions, and (b) if they are.
4. Assume that the number of messages input to a communication channel in an interval of duration t
seconds is Poisson distributed with parameter 0.3t. Compute the probabilities of the following
events:
(a) Exactly two messages will arrive during a five-second interval,
(b) At most eight messages arrive in a period of ten seconds,
(c) The number of message arrivals in an interval of duration twenty seconds is between eight and
twelve.
5. Consider a communication channel, where the probability of error-free transmission of a packet has
some fixed value p. Suppose further that if a packet is erroneous when received, a retransmission is
initiated. This is repeated until an error-free transmission occurs. Moreover, the outcome of each
transmission is independent of previous transmissions. To put this procedure in a more formal set-
