CHARACTERIZING DISCRETE DISTRIBUTIONS

When we deal with sampled data, it is customary to plot a histogram of relative frequencies in order to figure out how the data are distributed. When we consider a discrete random variable, we may use more or less the same concepts in order to provide a full characterization of uncertainty. For instance, if we consider dice throwing, a full description of the related random variable is given by

This is an example of a probability mass function, which we formally define below and fully characterizes the probability distribution of a discrete random variable. However strange it may be, it turns out that a more flexible characterization is obtained by referring to cumulative relative frequencies, which leads to the definition of a cumulative distribution function. The essential advantage is that the latter function is more general and translates directly to the case of continuous random variables. In the next two sections we explore these concepts in detail.