lar arrangement of pegs or nails makes it possible to repeat the ball's choice as many times as there are rows (n) in the matrix. At the bottom and after the final row, the ball rests in a single bin, which is associated with some reward or gain. It is easily established that the probability for the ball to be found in a given bin k follows the binomial distribution, Eq. (2.9). If the number n of rows becomes large, and for a sufficiently large number of such trials, the histogram distribution of balls into the bottom bins takes a Gaussian-like envelope, which represents a nice, mechanical schoolroom illustration of the CLT. This concludes the second chapter on probability basics. Most of the mathematical tools that are required to approach information theory have been described in these two chapters. 3 See, for instance: www.stat.sc.edu/~west/javahtml/CLT.html, www.rand.org/statistics/applets/clt.html, www.math.csusb.edu/faculty/stanton/m262/central_limit_theorem/clt_old.html, www.ruf.rice.edu/%7Elane/stat_sim/sampling_dist/index.html, www.vias.org/simulations/simusoft_cenlimit.html. 4 I am grateful to Professor Todd Ogden for permission to reproduce the simulation results obtained from his web tool in www.stat.sc.edu/~west/javahtml/CLT.html. 2.6 Exercises k = 10 Roll the dice 5 dice Number of rolls: k = 100 k = 1000 Roll the dice 5 dice Number of rolls: Roll the dice 5 dice Number of rolls: k = 10000 k = 100000 Roll the dice 5 dice Number of rolls: 10000 Roll the dice 5 dice Number of rolls: 10000 Figure 2.9 Proof of the central-limit theorem, through five-dice rolling experiment, (a) from k = 10 to k = 1000 trials, (b) from k = 10 000 to k = 100 000 trials, showing asymptotic convergence of histogram shape towards a Gaussian or normal distribution envelope. Exercises 2.1 (B): Calculate the mean and standard deviation of a discrete uniform distribution defined over the integer interval [1, 5]. 2.2 (B): A radioactive substance has a decay rate of 0.1 events per second. Taking a group of ten atoms, what is the pr