# Random variable

Summary

Definition (simplified)

- Given:
- An experiment
- A sample space \(S\) and events
- A probability measure \(P \) assigning a probability to each event. \(P\) satisfies the probability rules.
- A random variable \(X\) is a function with domain \(S\) .
- If the codomain is \(R\) then we say that \(X\) is a real-valued random variable.

Example

- Experiment that picks an outcome from \(S=\{ a,b,c \}\) . Each outcome is equally likely.
- 8 possible events.
- \(X:S→R\) defined by \(X\left( a \right)=1\) , \(X\left( b \right)=2\) and \(X\left( c \right)=3\) is then a (real-valued) random variable