University Problem Set #5 ECE 313
of Illinois Page 1 of 4 Fall 2001
Assigned: Wednesday, September 19
Due: Wednesday, September 26
Reading: Ross, Chapters 4 .8, 4.9, and Chapter 3
Noncredit Exercises:
Chapter 4 Problems: 49, 51, 53, 57-59, 70; Theoretical Exercises: 16-18, 25, 26.
Chapter 3 Problems: 1, 2, 5, 10, 12, 16, 31, 38, 39, 44; Exercises at the end of this problem set.
Problems:
1. The Sirrah Poll wishes to assess what percentage of voters believe Governor Shrub’s claim
that he was shocked, shocked to hear that the word Rats was used subliminally in one of
his TV commercials. A random sample of N voters is asked for opinions. Assume that
each voter decides the matter independently. Thus, the number of voters believing the
Governor can be modeled as a binomial random variable X with parameters (N,p) where p
is the probability that a randomly chosen voter believes the Governor. The Poll knows the
value of X that it obtained, and it wishes to estimate the value of p and report this.
(a) What is the maximum-likelihood estimate of p? Call this ^
p. Note that ^
p is a function of X.
(b) The Poll wants to be fairly sure that its estimate ^
p has a margin of error of at most 2%. The
Poll thus wants to have the following inequality hold:
P{|^
p – p| ≥ 0.02} ≤ 0.05.
The Chicago Tribune then reports on its front page that the Sirrah Poll had found that
100^
p% of the people believed Governor Shrub, and that the margin of error of the poll is
±2%. Use the Chebyshev inequality to determine the value of N that will guarantee a
confidence interval of width 0.04 with a confidence level of 95%.
2. [“Let your communication be Yea Yea, Nay Nay, for whatsoever is more than this cometh of evil”]
The bits transmitted over a communication link are in error with probability p independently
of each other. Since p is too large, a simple repetition code is used in which each data bit is
sent n times (i.e. sent repeatedly) over the link. Assume that n is odd. The receiver
examines the n received bits and decides that the transmitter was trying to send a data bit 0
or 1 according as the majority of the n received bits received is 0 or 1.
(a) Let X denote the number of errors in the n received bits corresponding to a data bit. What
kind of random variable is X? What is the mean of X? What is its variance?
(b) The receiver decision is in error if and only if the event {X > n/2} occurs. Express the
probability that the receiver decision is in error as a function of n and p, and compute its
numerical value for the case n = 3, p = 10–2. Compared to just transmitting the data bit
once and receiving it with probability of error p, does the repetition code reduce the
probability that the receiver decision is in error? Since there is no such thing as a free
lunch, what is being given up in thus reducing the bit error probability?
(c) You are asked to choose a value of n for the repetition code such that the probability that the
receiver decision is in error is at most 10–5. Estimate P{X > n/2} using the Chebyshev
inequality and use this to find the smallest value of n that meets the desired specification.
3. [“I’m leaving on a jet plane…”] Suppose that 105 passengers hold reservations for a 100–seat
flight from Chicago to Champaign. The number of passengers showing up for the flight
can be modeled as a binomial random variable X with parameters (105,0.9), that is, each is
deciding independently of the others (with probability 0.9) to show up. Note that this
simple model does not take into account things such as everyone in a family either all
showing up or all failing to show.
(a) Find the probability that all passengers who show up get seats, i.e. find P{X ≤ 100}.
(b) Explain why the number of no-shows can be modeled as a Poisson random variable Y, and
compute the value of the parameter λ.
(c) Compute the probability that all passengers who show up get seats based on this Poisson
model, i.e find P{Y ≥ 5}, and compare to the “more exact” answer of part (a)
