e for random variables x (top) and x2 (bottom). Counts -13 -10 -7 -4 -1 2 5 8 15 Z = 50x y = 100x Figure 2.8 Histogram of the 200 sampling points x shown in Fig. 2.7, as converted into y = 100x, and corresponding Gaussian distribution envelope p(y). The inset shows a denser histogram corresponding to the same x data with Z = 50x. Probability distributions reason, the envelope shape is closer to that of a bell. To obtain a smooth histogram envelope that would closely fit the Gaussian bell curve, one would need to acquire 104-105 sampling points and arrange them into hundreds of histogram bins. The lesson learnt is that experimental statistics require large numbers of samplings in order to reflect a given probability law, for instance, the Gaussian PDF (but it is not limited to this case). We have previously reached a similar conclusion from our earlier coin-flipping experiment, for which the associated probability distribution is discrete and quite elementary (p(heads) = p(tails) = 1/2), see Chapter 1 and Fig. 1.3. To recall, it took no less than 700 samples to approach the expected uniform distribution with reasonable accuracy. The Gaussian or normal probability distribution characterizes a large variety of ran- dom processes found in physics, in engineering, and in many other domains of science. In most random processes indeed, the uncertainty associated with continuous param- eters, which is also referred to as noise, obeys a Gaussian (normal) PDF. Here is a nonlimitative list of Gaussian (normal) processes: Experimental measurement errors (the mean <x> being taken as the value to be retained); Manufacturing, in the distribution of production yields and quality scoring; Telecommunications, in the distribution of 1/0 bit errors in digital receivers; Photonics, to approximate the transverse or spatial distribution of light intensity in optical fibers or in laser beams; Education and training, in the distribution of intelligence (IQ) test scores, or profes- sional qualifications and performance ratings; Informatio