LINKÖPINGS UNIVERSITET
Department of Computer and
Information Science
Division of Statistics/ANd
732A45 Statistical Evidence Evaluation
Fall semester 2015
Assignment 2
Assignment 2
Below are three tasks that you shall try to solve. All questions put should be answered.
Prepare your solutions in a nice format that can be easily read. There is also an additional
optional task for those who wish more challenges.
Your solutions should be submitted at latest on Friday 20 November 2015.
1.
The beta distribution (see Assignment 1 for the probability density function) is a
conjugate family of distribution for Bernoulli experiments, i.e. experiments with only
two outcomes (success and failure). A number of independent repeats of the same
Bernoulli experiment would give the binomial distribution for the number of repeats
with the same outcome. For a beta distribution with shape parameters a and b the mean
is a/(a+b) and the variance is ab/[(a+b)2(a+b+1)].
An experiment which gives a successful result with probability  is independently
repeated 20 times resulting in 6 successes. Suppose our previous experience tells us that
the probability of obtaining a successful result is around 20 %. Use this information to
obtain a Bayesian point estimate of  that is the optimal action with respect to quadratic
loss.
2.
Assume that if an individual is a resident of Australia they would with probability 0.75
speak English with a typical Australian accent. Among individuals that are not residents
of Australia the probability they will speak English with an Australian accent is 0.001.
Moreover, assume that if an individual speaks English with a typical Australian accent
they would now and then use a specific diphthong D with probability 0.3 (and not use it
at all with probability 0.7). This diphthong is hardly used by someone who does not
speak English with a typical Australian accent. We can assume this probability to be
one out of 10 000.
Construct a Bayesian network from which you can calculate the conditional probability
that an individual is a resident of Australia given that they now and the use the
diphthong D when speaking.
3.
Assume that a governmental building gets a telephone call with a bomb scare. Upon
that the manager(s) will decide whether the whole building or part of it should be
evacuated. The additional cost of a total evacuation (compared to a partial one) is
estimated to be around US$ 50 000. However if there really is a bomb explosion
according to the threat several people may be injured and there will be a lot of material
damages. The additional cost of this scenario is estimated to reach US$ 10 million
(taking personal suffering for longer periods into account).
LINKÖPINGS UNIVERSITET
Department of Computer and
Information Science
Division of Statistics/ANd
732A45 Statistical Evidence Evaluation
Fall semester 2015
Assignment 2
Real bomb explosions are not at all common in this part of the country. One estimates
that the probability there will be a bomb explosion at this time is about 0.00001. If a
bomb is placed there is a 99.5 percent chance that there will be a bomb threat (like the
one was received), but there are a lot of false alarms. In ten of the previous 200 days
there were 10 bomb scares, but in all these cases there was no bomb. The 99.5 percent
figure is based on more information than pure statistics.
What is the optimal decision to be made by the managers, total or partial evacuation, if
they should minimise the expected loss? Construct an influence diagram for this
problem from which you can do the calculations.
4.
*(Optional task for those of you who would like a challenge)
Assume you are about to drive a distance of 100 km. If you manage to do this within 1
hour and 6 minutes you will get a reward of US$ 5000. If the time needed is longer you
will not get any reward.
The speed limit is 90 km/h the whole distance (hence there are no crossings where you
will have to stop to give way for other cars. However with 100 % probability you will
be objected to a speed control (but you don’t know where), and if you exceed the speed
limit you will be fined with an amount that is US$ 1000 times the square of the excess
of the speed limit, i.e. if you have the limit 92 km/h you will be fined with US$
1000×(92-90)2 = US$ 4000.
Assume your car is equipped with a cruise control that you set to a certain speed. In
reality your speed will be uniformly distributed (a bit too ideal, but still) ± 2 km/h
around the nominal speed set. For instance, if you set the speed to 90 km/h your actual
speed will be U(88,92).
What would be an optimal setting of the speed if you wish to maximise your expected
utility for this trip?