University Problem Set 8 ECE 313
of Illinois Fall 1998
Assigned : Friday, October 16, 1998
Due : Wednesday, October 21, 1998
Reading : Ross, Chapter 4
Recommended additional problems : 4.1, 4.3, 4.4, 4.7–4.10, 4.12–4.16, 4.19, 4.39, 4.40, 4.42–4.44, 4.47–
4.49, 4.51–4.54, 4.61–4.64, 4.66, 4.69, 4.70, 4.73, 4.76, 4.77
1. Ross, p. 180, Problem 45.
2. (a) In Problem 3 of Homework#5 you calculated the probability that all passengers, out of
75
, that
showed up for a
70
–seat flight got seats. This was done via the binomial theorem. If we now let
X
be
a random variable denoting the number of passengers that do not show up for the flight, what kind of
random variable is
X
, and what are its parameters? Now express the probability that everyone who
shows up gets a seat in terms of theCDF or pmf of
X
.
(b) We can also calculate the above probability (refer to HW#5 solutions) using a Possion approximation
to the above random variable
X
.Let
Y
be a Poisson random variable with parameter
=75
0
:
2=
15
. Use the Poisson pmf to evaluate the probability that all passengers who show up for the flight get
seats. Compare this number to the actual probability and find the relative error.
(c) Ross, p. 181, Problem 58.
The binomial random variable with parameters
(
n; p
)
is reasonably well approximated by a Poisson random
variable with parameter
=
np
,when
n
becomes very large and
p
becomes small. That is, under the above
conditions, the two pmfs are approximately equal (You can try to prove this yourself or look into Ross, p.154
for a proof).
3. The Poisson random variable can be used to capture the random behavior of many physical phenomenon:
for instance, the incidence of telephone calls at a switching station. Suppose that telephone calls coming
into a switching station obey a Poisson probability law at a rate of
12
a minute.
(a) What is the probability that, during a
5
-second period, either no call or at least
2
calls arrive at the
switchboard?
(b) If you observed
12
five-second intervals, would you be surprised if
8
or more of these had either no
call in them or at least
2
calls in them?
(c) Poorly modeled call traffic into a switchboard can lead to calls being “dropped” once in a while. This
is obviously something that service providers try to avoid as much as possible to maximize revenue
(can you imagine AT&T going out of their way to help you?!). If the switchboard is designed to not
drop more than
5%
of the incoming calls in a minute, what should it’s capacity be (that is, how many
calls should it be capable of handling a minute)?
4. Ross, p. 182, Problem 65.
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