Due on the day of the final exam
May be done as a team project with three or fewer team members.
Each team member must submit an individual report.
Problem 1
A user has four cell phones, some of them in location A and the
others in location B. The user moves between A and B and takes
one of the phones only if he takes a taxi to change his location,
but does not take a phone if he takes his car. Let p be the
probability that the user takes a taxi.
1. Find the probability that no phone is available to him when
he changes location.
2. For a value of p = 0.6, find the number of phones that would
be required for the probability of not having a phone available
when the user changes location to be less than 0.1.
Problem 2
A game starts with 3 coins in a box. At each turn the number of
coins in the box is counted and the following procedure is
repeated k times:
A fair die is thrown, and depending to the outcome the following
four things can happen:
If the outcome is 1 or 2 the player takes 1 coin from the
box.
If the outcome is 3 no action is taken.
If the outcome is 4 the player puts 1 coin in the box
(assume that the player has an unlimited supply of coins).
If the outcome is 5 or 6, the player puts 2 coins in the
box.
Whenever the box is empty, the game stops.
1. Compute the expected number of coins in the box after turn n
2. Compute the probability that the game will stop eventually.
Problem 3
A signal X(t) with power spectral density S(w) = 2aP/(w2 + a2 )
and additive noise N(t) with one-sided power spectral density N0
are passed through a filter with frequency response H(w) = A/(jw
+ b).
1. Find the signal power and noise power at the filter output.
2. Find the filter coefficients A and b that maximize the signal-
to-noise ratio.
Problem 4
Given a binomial random variable K with probability of success p
and n trials:
It is known that for E(K) not very small or very large the
distribution can be approximated by a Gaussian distribution.
It is also known that for E(K) small the distribution can be
approximated by a Poisson distribution.
1. Compare the Binomial and Poisson distributions for equal
expected values and for k=3, n=100 and p=0.05. and for k=40,
n=100 and p=0.45. For these values of the parameters compute
the percent difference between the two distributions.
2. Compare the Binomial and Gaussian distributions for equal
expected values and variances for k=40, n=100 and p=0.45. For
these values of the parameters compute the percent difference
between the two distributions.