# Random variable

Outcomes describe the qualitative aspect of an experiment: what happened? Often, after performing a probabilistic experiment, we want to measure various quantitative aspects of the outcome and relate them to each other.

Random variables are the technical tool we use for this. A random variable gives a numeric value to each outcome:

Definition: Random variable
A (real-valued) random variable on a probability space is a function . More generally, if is any set, an -valued random variable is a function .
• For example, if we were to model a game where I roll a die, and I win \\$10 if I roll a 6 and lose \\$3 if I roll 3 or less, then a reasonable sample space would be the set , and the winnings would be described by a random variable given by , , and .
• For example, if we were to model an experiment where I select a person and sample their height, then a reasonable sample space would be the set of people, and the random variable of interest would be the function where is the height of person .