This module covers the Geometric and Poisson distributions you find in Chapters 2.2 and 2.4.

The geometric distribution was introduced as the number of trials before the first success when you have independent Bernoulli trials. The geometric distribution plays another important role in modeling: it is the “maximum entropy” distribution among all distributions over natural numbers with a given mean. While this course is too premature to introduce the “maximum entropy” concept formally, the intuitive meaning is that it is the distribution that makes the least assumptions.

The Poisson was introduced as a limiting case of Binomial random variables when the number of trials goes to infinity, but the success paramater diminishes to zero in just the right way. Needless to say, the Poisson model is also extremely important in modeling, It shows up in another way when analyzing and simulating randomized algorithms, by means of what is known as the Poissonization approach.

When you take \(n\) independent Bernoulli trials and record the number of ones (say \(N_1\)) and zeros (say \(N_0\)), the random variables \(N_1\) and \(N_0\) are not independent (clearly, since \(N_1=n-N_0\)). But suppose you took a different perspective on this, and treated \(n\) not as a fixed number by a random variable itself. So you would first choose a value for \(N\) according to the probability model, and then do \(N\) independent Bernoulli trials and record the number of 1s and 0s in \(N_1\) and \(N_0\) respectively. So conditioned on \(N\), \(N_1\) and \(N_0\) are not independent for the same reason as above. But if \(N\) is chosen to be Poisson, then \(N_0\) and \(N_1\) turn out to be independent Poisson random variables as well! We will look at this in one of the Problems in class.