Discrete Random Variables

Random variables provide a more compact description of an experiment than the finest grain description of the sample space. The notion of random variables provides us the power of abstraction and thus allows us to discard unimportant details in the outcome of an experiment. Virtually all serious probabilistic computations are performed in terms of random variables. A random variable defined on a discrete sample space will be discrete, while it is possible to define a discrete random variable on a continuous sample space. The cumulative distribution function contains most of the interesting information about the underlying probability system and will be used extensively. The notion of probability generating functions (PGFs) is a convenient tool that simplifies computations involving integer-valued, discrete random variables. This chapter explains that the problem of determining the compound probability mass function (pmf) given the marginal pmf's does not have a unique solution, unless the random variables are independent.