 the square of the mean. It is always a positive number. We can thus take its square root, which is called the PDF standard deviation: =>-<x>2 (2.4) How does one calculate the mean square <x2>? Consistent with the earlier definition in Eq. (2.1), we have (2.5) Probability distributions 0.180 0.160 0.140 0.120 0.100 0.080 0.060 0.040 0.020 0.000 Count Figure 2.1 Same probability distribution function (PDF) as in Fig. 1.2(b), illustrating the concepts of mean (<x> =7) and standard deviation (o = 2.415). As an illustration, let us then calculate the standard deviations of the one-die and two-dice PDF. In the one-die case, we have 15.166. (2.6) The one-die PDF variance is, thus, o2 <x2>-<x>2=15.166-3 - 3.52 = 2.916, and the standard deviation 1.707. We can get a better sense of the meaning of such a "deviation" with the two-dice example. It takes a computer spreadsheet (or pocket-calculator patience) to com- pute <x2> with the PDF shown in Fig. 1.2(b), but the reader will easily check that the result is <x2> = 54.833, hence o2 <x2>. <x>2 = 54.833 - 72 = 5.833, and = 5583 Reproducing, for convenience, the same PDF plot in Fig. 2.1, we see that the interval 2 [4.5, 9.5], of width 20, defines the PDF region where the probabilities are the highest, i.e., about greater or equal to half the peak value. In other words, the outcome of the two-dice roll is most highly likely to fall within the 20 region. Statistics experts familiarly state that the event to be expected is <x> with a to accuracy or trust interval. However, such a conclusion does not apply with other types of PDF, for instance, with the uniform distribution, as I will show further below. Exponential, Poisson, and binomial distributions I shall now consider three basic examples of discrete PDFs. As we shall see, these PDFs govern many physical phenomena and even human society! They are the 2.2 Exponential, Poisson, and binomial distributions Figure 2.2 Plots of discrete-exponential or Bose-Einstein probability distribution corresponding to mean values <n> = N = 