n theory, which will be developed in Chapter 4 when analyzing continuous channels with noise. Medicine, in the distribution of blood pressure and hair length, or of the logarithm of body weight or height; Economics and finance, in the distribution of the logarithm of interest rates, exchange rates, stock returns, and inflation. Processes that are associated with a Gaussian (normal) distribution are said to respond to normality. If normality is satisfied only with the logarithm of the variable x (as seen in the last two above examples), the process is said to be log-normal. By way of simplified explanation, normality comes from the additive effect of independent random factors, while log-normality comes from their multiplicative effect. Generally, the Gaussian (normal) law represents a good approximation of most con- tinuous distributions, provided the number of events or samples is relatively large. This will be shown in the next section. Furthermore, the Gaussian (normal) law represents the asymptotic limit of most discrete probability distributions with large mean <k>, two key examples being the binomial and the Poisson PDFs described in the previous section. As we have seen, the binomial distribution Pk = Ckqk(1 - q)n-k, which is For advanced reference, the log-normal distribution is defined as: plx)= mean and variance are N = exp(u +s2/2) and o2 = [exp - respectively. 2.5 Central-limit theorem defined for integers k=0,..., n, Eq. (2.9) converges towards the Poisson distribution, Eq. (2.8) in the limit of large N. In the same limit, it can be shown that both distri- butions converge towards a Gaussian (normal) PDF of same mean <k> and variance o2=nq(1-q). Central-limit theorem In probability theory, there exist several central-limit theorems, which show that the sum of large numbers of independent random variables, each having a different probability distribution, asymptotically converges towards some kind of limiting PDF. Remarkably, this limiting PDF is the same, regardless of the initial distribution of the