2.6 Discrete & Continuous Random Variables

What, if anything is fundamentally different between discrete and continuous random variables? As a way of starting the conversation, consider the following cases:

• Suppose \(X\) is a random variable that describes the time a student spends on w203 homework 1.
  • If you have only granular measurement – i.e. the number of nights spent working on the homework – is this discrete or continuous?
  • If you have the number of hours, is it discrete or continuous?
  • If you have the number of seconds? Or milliseconds?
• Is it possible that \(P(X = a) = 0\) for every point \(a\)? For example, that \(P(X = 3600) = 0\).
• Does one of these measures have more information in it than another?
  • How are measurement choices that we make as designers of information capture systems – i.e. the machine processes, human processes, or other processes that we are going to work with as data scientists – reflected in both the amount of information that is gathered, the type of information that is gathered, and the types of random variables that are manifest as a result?