What defines a random variable?

A random variable is fundamentally defined as a variable that can take on different numerical values based on the outcomes of a random experiment. This concept is crucial in probability and statistics, where the outcomes of experiments are uncertain, and the random variable quantifies these outcomes.

In practical terms, a random variable can be either discrete, taking on a countable number of values, or continuous, taking on an infinite number of possible values within a given range. For instance, if you roll a die, the random variable could represent the number that comes up, which is determined by the random event of rolling that die.

This definition makes it clear that the values of random variables are inherently linked to the randomness of the underlying experiment, which differentiates them from constants or fixed values and relates them directly to real-world applications in statistical analysis and probability theory.