1, 2, 3, 4, 5 (open symbols corresponding to the case <n> = 1). The inset shows the same plots in decimal logarithmic scale. discrete-exponential distribution, the Poisson distribution, and the binomial distribu- tion. To recall that these three PDFs concern discrete random variables, I will use the notations <n> and p(n), instead of <x> and p(x), which we reserve for continuous distributions. The discrete-exponential distribution, also referred to as the Bose-Einstein (BE) distribution, is defined according to (2.7) where <n> = N is the mean value. This PDF variance is simply o2=N+N2. Figure 2.2 shows plots of the exponential distribution for various values of N, in both linear and logarithmic scales. The continuous lines linking the data are only shown to guide the eye, recalling that the PDF applies to the discrete variable N. The PDF is seen to be linear in the logarithmic plot, which allows a better visualization of the evanescent tail at high n. As the figure also illustrates, the peak value of the exponential/BE distribution is reached at the origin n = 0, with p(0) = /(N + 1). In the physics world, such a distribution is representative of thermal processes. Such processes concern, for instance, the emission of photons (the electromagnetic energy quanta) by hot sources, such as an incandescent light bulb or a star, like the sun, or the distribution of electrons in atomic energy levels, and many other physical phenomena. As we shall see in the forthcoming chapters, the discrete-exponential distribution is important in information theory. We will also see that the exponential PDF is commonly found in human society, for instance, concerning the distribution of alphabetic characters in Western languages. Let us consider next another PDF of interest, which is the Poisson distribution. This PDF is used to predict the number of occurrences of a discrete event over a fixed Probability distributions Figure 2.3 Plots of the Poisson distribution corresponding to mean values N = 1, 3, 5, 7, 9 (open symbols correspondi