2.4 Using Definitions of Random Variables

2.4.1 Random Varaible

What is a random variable? Does this definition help you?

Definition 2.1 (Random Variable) A random variable is a function \(X : \Omega \rightarrow \mathbb{R},\) such that \(\forall r \in \mathbb{R}, \{\omega \in \Omega: X(\omega) \leq r\} \in S\).

Someone, please, read that without using a single “omega”, \(\mathbb{R}\), or other jargon terminology. Instead, someone read this aloud and tell us what each of the concepts mean.

The goal of writing with math symbols like this is to be absolutely clear what concepts the author does and does not mean to invoke when they write a definition or a theorem. In a very real sense, this is a language that has specific meaning attached to specific symbols; there is a correspondence between the mathematical language and each of our home languages, but exactly what the relationship is needs to be defined into each student’s home language.

• What are the key things that random variables allow you to accomplish?
  • Suppose that you were going to try to make a model that predicts the probability of winning “big money” on a slot machine. Big money might be that you get :cherries: :cherries: :cherries:. Can you do math with :cherries:?
  • Suppose that you wanted to build a chatbort that uses a language model so that you don’t have to do your homework anymore. How would you go about it?
  • Suppose you want to direct class support to students in 203, but their grades are scored [A, A-, ..., ] and features include prior statistics classes grades, also scored A, A-, ...]