The shift from discrete to continuous random variables

Consider the function $f: \mathbb{N} \to \mathbb{R}$ where

$$ f(x) = x^2 $$

This is a simple quadratic equation you’re acquainted with, but note that our domain are only natural numbers, meaning $1, 2, 3, 4, \ldots$. The graph of this function looks like this:

But this may not be the graph of the most common version of this quadratic. Indeed, it’s more commonly defined as a function from $f: \mathbb{R} \to \mathbb{R}$. While the natural numbers are said to be countable, the real numbers are uncountable.

Intuitively, if you go about counting every natural number, even though they go from $1$ to infinity, you can still attempt to enumerate them starting from $1, 2, 3, 4, 5, \ldots$. It will take you forever, but at least you can count them.

But try to count every real number from $0$ to $1$. You have $0$, then $0.1$, but wait—you have $0.01$, then $0.001$, then $0.0001$, then $0.00001$… See how many zeros you’re adding? You won’t even reach $0.2$ because between any two numbers you pick, there are infinitely many more numbers! The reals are said to be uncountable.

From finite sample spaces to continuous ones

Just as the quadratic function’s domain isn’t limited to countable sets like the natural numbers, probability functions can also take an uncountable sample space $\Omega$ as opposed to discrete ones like that of a coin flip.

Remember that probability is a function from the event space to real numbers between $0$ and $1$ (see [axioms of probability]). When the sample space was a coin flip, the set of all possible events (the event space) is finite:

$$ \Omega = \{\text{Heads}, \text{Tails}\} $$

You can count and list every possible outcome.

Real-world example: Human weight

But we’re not limited to countable sample spaces, not even in real life. Certain outcomes are not countable. The mass that human beings can have can be anywhere in a range.

The lowest weight of a baby ever recorded was approximately $0.24$ kg (the baby born in 2018 in San Diego), and the heaviest adult recorded was approximately $635$ kg. So human weight can be anywhere between those two values. You can be $0.24$ kg or $635$ kg. The possible outcomes lie in the interval $[0.24, 635]$. This isn’t a countable sample space, just like the interval of all reals from $[0, 1]$ wasn’t. If we want to find the probability of a human being having a certain weight, the sample space will be our uncountable interval.

If a sample space is uncountable, then we have a continuous random variable (see random variables).

Exercise: Distinguish discrete from continuous

Identify whether each random variable has a discrete or continuous sample space:

1. The number of heads in $10$ coin flips
2. The exact time a bus arrives (in seconds)
3. The number of students in a classroom
4. The temperature of water in degrees Celsius
5. The number of cars passing through an intersection