ng to the case <n> = 1). time interval. If N is the expected number of occurrences over that time interval, the probability that the count is exactly n is p(n) = (2.8) As a key property, the Poisson PDF variance is equal to the PDF mean, i.e., o2 ==N. Figure 2.3 shows plots of the Poisson PDF for various values of the mean N. It is seen from the figure that as N increases, the distribution widens (but slowly, according to o = NV. Also, the line joining the discrete points progressively takes the shape of a symmetric bell curve centered about the mean, as illustrated for N = 9 (the mean coinciding with the PDF peak only for N 80). This property will be discussed further on, when considering continuous distributions. There exist numerous examples in the physical world of the Poisson distribution, referred to as Poisson processes. In atomic physics, for instance, the Poisson PDF defines the count of nuclei decaying by radioactivity over a period t. Given the decay rate a (number of decays per second), the mean count is N = At, and the Poisson PDF p(n) gives the probability of counting n decays over that period. In laser physics, the Poisson PDF corresponds to the count of photons emitted by a coherent light source, or laser. Poisson processes are also found at macroscopic and human scales: for instance, the number of cars passing under a highway bridge over a single lane, the number of phone calls handled by a central office, the number of Internet hits received by a website, the number of bonds traded in the stock exchange, or the number of raindrops falling on a roof window. Each of these counts, being made within a given amount of time, from seconds to minutes to hours to days, obeys Poisson statistics. The binomial distribution, also called the Bernoulli distribution, describes the number of successes recorded in a discrete sequence of independent trials. Here, let's call k the random variable, which is the number of successes, and n the fixed number of trials, with k < n. If the probability of success for a sin