## What is a random variable?

## Answer

The formal mathematical definition of a random variable is “a function from a sample space to real numbers.” This definition is concise, but also can be confusing.

Start by defining the sample space as all possible outcomes of an experiment. Suppose a coin is flipped two times. The sample space would be {HH, TH, HT, TT} where H represents heads and T represents tails. The sample space by itself is difficult to analyze because we don’t have any real numbers to work with. This is why random variables are used; they create a numerical outcome for the experiment.

For the experiment above, we could assign random variables as X = the number of heads or Y = the number of tails. Both random variables can take the values 0, 1, or 2. The random variable is the function that allows us to take the outcomes of the experiment and describe a relevant numerical result.

Random variables are denoted with upper case letters (X, Y, Z, etc.) and the values the random variables assume are denoted with lowercase letters (x, y, z, etc.).