19.1 Random variables
Hold your horses, though; it’s not that simple. Random variables are hard to understand in their general form, so we’ll slow down and focus on special cases, taking one step at a time. This is how learning is done most effectively, and we’ll follow this path as well.
Let’s deal with so-called discrete random variables (such as the above example) first, real random variables second, and the general case last.
19.1.1 Discrete random variables
Following our motivating example describing the number of heads in n coin tosses, we can create a formal definition.
Definition 80. (Discrete random variables)
Let (Ω,Σ,P) be a probability space and {xk}k=1∞ be an arbitrary sequence of real numbers. The function X : Ω → {x1,x2,…} is called a discrete random variable if the sets
are events for any integer k ∈ℤ (that is, Sk ∈ Σ).
You might ask why we are requiring the sets {&...
