Discrete and Continuous Random Variables

A random variable can behave in one of two fundamentally different ways, depending on the kind of values it can take.

A discrete random variable takes distinct, countable values with no possible value existing in between two consecutive ones — much like an hour-and-minutes-only watch display that jumps straight from 12:00 to 12:01 with nothing shown in between. Tossing a single coin and recording heads as $0$ and tails as $1$ is another example: the outcome is always one of these two isolated values, never something in between. Every possible value of a discrete random variable can be paired with a non-zero probability of occurring.

A continuous random variable, by contrast, has its values obtained by measurement rather than counting, and can take any of infinitely many values within a range — like a watch that also displays seconds, where the elapsed time between two readings can itself be any value in between. Because its possible values form an uncountable, infinite set (called the range of the variable), a continuous random variable cannot be listed out value by value the way a discrete one can. …