19.2 Discrete distributions

Let’s recap what we have learned so far. In probability theory, our goal is to first model real-life scenarios affected by uncertainty and then to analyze them using mathematical tools such as calculus.

For the latter purpose, probability spaces are not easy to work with. A probability measure is a function defined on an σ-algebra, so we can’t really use calculus there.

Random variables bring us one step closer to the solution, but they can also be difficult to work with. Even though a real-valued random variable X : Ω →ℝ maps an abstract probablity space to the set of real numbers, there are some complications. Ω can be anything, and if you recall, we might not even have a tractable formula for X.

For example, if X denotes the lifetime of a lightbulb, we don’t have a formula. So, again, we can’t use calculus. However, there is a way to represent the information contained by a random variable in a...