# Statistics: Discrete and Continuous Random Variables

In statistics, numerical random variables represent counts and measurements. They come in two different flavors: discrete and continuous, depending on the type of outcomes that are possible:

**Discrete random variables.**If the possible outcomes of a random variable can be listed out using a finite (or countably infinite) set of single numbers (for example, {0, 1, 2 . . . , 10}; or {-3, -2.75, 0, 1.5}; or {10, 20, 30, 40, 50…} ), then the random variable is*discrete*.**Continuous random variables.**If the possible outcomes of a random variable can only be described using an interval of real numbers (for example, all real numbers from zero to ten ), then the random variable is*continuous*.

Discrete random variables typically represent counts — for example, the number of people who voted yes for a smoking ban out of a random sample of 100 people (possible values are 0, 1, 2, . . . , 100); or the number of accidents at a certain intersection over one year's time (possible values are 0, 1, 2, . . .).

Discrete random variables have two classes: finite and countably infinite. A discrete random variable is *finite* if its list of possible values has a fixed (finite) number of elements in it (for example, the number of smoking ban supporters in a random sample of 100 voters has to be between 0 and 100). One very common finite random variable is obtained from the binomial distribution.

A discrete random variable is *countably infinite* if its possible values can be specifically listed out but they have no specific end. For example, the number of accidents occurring at a certain intersection over a 10-year period can take on possible values: 0, 1, 2, . . . (in theory, the number of accidents can take on infinitely many values.).

Continuous random variables typically represent measurements, such as time to complete a task (for example 1 minute 10 seconds, 1 minute 20 seconds, and so on) or the weight of a newborn. What separates continuous random variables from discrete ones is that they are *uncountably infinite;* they have too many possible values to list out or to count and/or they can be measured to a high level of precision (such as the level of smog in the air in Los Angeles on a given day, measured in parts per million).