Exercises to Data Mining course, 2005
  1. 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).
  2. 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.
  3. 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