m/CentralLimitTheorem.htn Probability distributions applets. 3 I consider here an experimental example of CLT proof, using one such web tool. Rolling dice and adding spots This is an experiment similar to that described earlier in Chapter 1 and illustrated in Figs. 1.1 and 1.2. As we know, each individual die has a uniform discrete distribution defined by p(x) = 1/6 with Xi = 1, 2, 3, 4, 5, 6 being the event space. If we roll two dice and sum up the spots, the event space is Xi = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and the probability distribution has a triangular (or witch's hat) shape centered about <x> = 7, see Fig. 1.2. Interestingly, the CLT does not apply to the two-dice case, the limiting PDF being still the witch hat, as can be easily verified. As a simplified explanation, this is because the event space is too limited. But when using three dice or more, the CLT is observed to apply, as illustrated in Fig. 2.9. The figure shows results obtained with five dice for k = 10, 100, 1000, 10 000, and 100 000. It is seen that as k increases, the resulting histogram envelope progressively takes a bell shape. Ultimately, the histogram takes a symmetrical (discretized) bell shape limited to the 26 = 30 5 + 1 integer bins of the event space. A 1000-dice experiment with an adequate number of trials k would yield a similar histogram with 5001 = 6000 - 1000 + 1 discrete bins, which is much closer to the idea of a smooth envelope, albeit the resulting PDF is discrete, not continuous. Only an infinite number of dice and an infinite number of rolls could provide a histogram match of the limiting Gaussian (normal) envelope. To complete the illustration of the CLT, consider the school game of a pegboard matrix, also known as a pinball machine, bean machine, quincunx, or Galton box, and whose principle is at the root of the Japanese gambling parlors called Pachinko. At each step of the game, a ball bounces on a peg (or a nail) to choose a left or right path randomly, according to a uniform, two-valued distribution. The triangu