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Mathematics of Machine Learning
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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.
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 {&...