random variable

It is a function that maps elements from a state space to $\mathbb{R}$. Random variables are written with upper case letters such as X, Y or Z. It is possible to map two elements from the state space to the same output.

The output of a random variable can be in a finite or infinite range. In both cases, we have two types:

Discrete

We could count the output elements of the random variable. That is, there are no values in between one element and the next (e.g. $\mathbb{N}$).

Continuos

The output elements could take any (infinite) values within a range (e.g. $\mathbb{R}$).

Why use random variables?

• Quantification: They map qualitative outcomes (e.g. “Sunny”) to real numbers ($\mathbb{R}$).

• Simplification: They allow us to ignore irrelevant details. If you only care about the sum of two rolled dice, a random variable $X$ treats $(2,4)$ and $(3,3)$ as the same value ($6$), even though they are different states.

• Functional Power: Once outcomes are numbers, we can use better use mathematical tools to describe the system behavior.

Overall, I think one of the strongest reasons why random variables are so useful is because of their abstraction power. We can say “this process follows a Normal Distribution” without needing to know if the underlying states are heights of people, errors in a clock, or light particles. It provides a universal language for different phenomena.