Exercises to Data Mining course, 2005
 Two points are selected uniformly and independently on
a 3D sphere
of radius 1.
(i) Find the distribution of the (3D Euclidean) distance
between the points.
(ii) Find the maximum probability, mean and
median distances between the points.
(iii) What are the MAP, mean and median estimates for the squared distance?
Hint: use a pdf whose variable is the squared distance.
(Note: will be discussed on Feb 4).

A sample y1,... yn of real numbers
has been obtained. It is known to be independent variables
with a common normal distribution. This distribution has unknown variance
s2 and the mean is known to be either 1 or 2, both cases considered
equally plausible.
(i) Describe a reasonable Bayesian method for deciding the mean value(1 or 2).
(ii) Characterize the power of the suggested procedure as a function of s2,
assuming that the sample consists of a single point.

Compute (using Matlab and Monte Carlo) an approximation to Pi
by estimating the probability that a needle, thrown on an area
covered by parallell lines, hits a line. The length of the needle
is the same as the distance between the parallell lines
(Buffons needle problem). Investigate how (and if) precision improves
with the number of throws performed.
Hint: the probability of hitting a line is 2/Pi